Publications of Jian-Guo Liu    :chronological  alphabetical  combined listing:

%% Books   
@book{fds165493,
   Title = {Multi-scale phenomena in complex fluids, Modeling, Analysis
             and Numerical Simulations},
   Publisher = {World Scientific},
   Editor = {T. Hou and C. Liu and J.-G. Liu},
   Year = {2009},
   ISBN = {978-981-4273-25-1},
   Key = {fds165493}
}

@book{fds165494,
   Title = {Hyperbolic Problems: Theory, Numerics and Applications,
             volume I: Plenary & Invited Talks; volume II: Contributed
             Talks},
   Volume = {67},
   Series = {Proceedings of Symposia in Applied Mathematics},
   Publisher = {American Mathematical Society},
   Editor = {E. Tadmor and J.-G. Liu and A.E. Tzavaras},
   Year = {2009},
   ISBN = {978-0-8218-4728-2},
   Key = {fds165494}
}

@book{fds70657,
   Title = {Dynamics in Models of Coarsening, Coagulation, Condensation
             and Quantization},
   Publisher = {World Scientific},
   Editor = {W. Bao and J.-G. Liu},
   Year = {2007},
   ISBN = {9789812708502},
   Key = {fds70657}
}


%% Papers Published   
@article{fds329519,
   Author = {Li, L and Liu, J-G and Lu, J},
   Title = {Fractional Stochastic Differential Equations Satisfying
             Fluctuation-Dissipation Theorem},
   Journal = {Journal of Statistical Physics},
   Volume = {169},
   Number = {2},
   Pages = {316-339},
   Year = {2017},
   Month = {October},
   url = {http://dx.doi.org/10.1007/s10955-017-1866-z},
   Doi = {10.1007/s10955-017-1866-z},
   Key = {fds329519}
}

@article{fds329520,
   Author = {Liu, J-G and Ma, Z and Zhou, Z},
   Title = {Explicit and Implicit TVD Schemes for Conservation Laws with
             Caputo Derivatives},
   Journal = {Journal of Scientific Computing},
   Volume = {72},
   Number = {1},
   Pages = {291-313},
   Year = {2017},
   Month = {July},
   url = {http://dx.doi.org/10.1007/s10915-017-0356-4},
   Doi = {10.1007/s10915-017-0356-4},
   Key = {fds329520}
}

@article{fds329521,
   Author = {Gao, Y and Ji, H and Liu, J-G and Witelski, TP},
   Title = {Global existence of solutions to a tear film model with
             locally elevated evaporation rates},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {350},
   Pages = {13-25},
   Year = {2017},
   Month = {July},
   url = {http://dx.doi.org/10.1016/j.physd.2017.03.005},
   Doi = {10.1016/j.physd.2017.03.005},
   Key = {fds329521}
}

@article{fds329522,
   Author = {Gao, Y and Liu, J-G and Lu, J},
   Title = {Continuum Limit of a Mesoscopic Model with Elasticity of
             Step Motion on Vicinal Surfaces},
   Journal = {Journal of Nonlinear Science},
   Volume = {27},
   Number = {3},
   Pages = {873-926},
   Year = {2017},
   Month = {June},
   url = {http://dx.doi.org/10.1007/s00332-016-9354-1},
   Doi = {10.1007/s00332-016-9354-1},
   Key = {fds329522}
}

@article{fds329524,
   Author = {Gao, Y and Liu, J-G and Lu, J},
   Title = {Weak Solution of a Continuum Model For Vicinal Surface in
             The Attachment-Detachment-Limited Regime},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {49},
   Number = {3},
   Pages = {1705-1731},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1094543},
   Doi = {10.1137/16M1094543},
   Key = {fds329524}
}

@article{fds329523,
   Author = {Huang, H and Liu, JG},
   Title = {Discrete-in-time random particle blob method for the
             Keller-Segel equation and convergence analysis},
   Journal = {Communications in Mathematical Sciences},
   Volume = {15},
   Number = {7},
   Pages = {1821-1842},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2017.v15.n7.a2},
   Abstract = {© 2017 International Press. We establish an error estimate
             of a discrete-in-time random particle blob method for the
             Keller{Segel (KS) equation in ℝ d (d≥2). With a blob
             size ε=N -1/d(d+1) log(N), we prove the convergence rate
             between the solution to the KS equation and the empirical
             measure of the random particle method under L 2 norm in
             probability, where N is the number of the
             particles.},
   Doi = {10.4310/CMS.2017.v15.n7.a2},
   Key = {fds329523}
}

@article{fds323838,
   Author = {Degond, P and Liu, J-G and Merino-Aceituno, S and Tardiveau,
             T},
   Title = {Continuum dynamics of the intention field under weakly
             cohesive social interaction},
   Journal = {Mathematical Models & Methods in Applied
             Sciences},
   Volume = {27},
   Number = {01},
   Pages = {159-182},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1142/S021820251740005X},
   Doi = {10.1142/S021820251740005X},
   Key = {fds323838}
}

@article{fds329525,
   Author = {Gao, Y and Liu, J-G},
   Title = {Global Convergence of a Sticky Particle Method for the
             Modified Camassa--Holm Equation},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {49},
   Number = {2},
   Pages = {1267-1294},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1102069},
   Doi = {10.1137/16M1102069},
   Key = {fds329525}
}

@article{fds320659,
   Author = {J.-G. Liu and J. Wang},
   Title = {A generalized Sz. Nagy inequality in higher dimensions and
             the critical thin film equation},
   Journal = {Nonlinearity},
   Volume = {30},
   Pages = {35-60},
   Year = {2017},
   Key = {fds320659}
}

@article{fds329169,
   Author = {Liu, J-G and Cong, W},
   Title = {Uniform $L^{\infty}$ boundedness for a degenerate
             parabolic-parabolic Keller-Segel model},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {22},
   Number = {2},
   Pages = {307-338},
   Year = {2016},
   Month = {December},
   url = {http://dx.doi.org/10.3934/dcdsb.2017015},
   Doi = {10.3934/dcdsb.2017015},
   Key = {fds329169}
}

@article{fds318453,
   Author = {Huang, H and Liu, J-G},
   Title = {A note on Monge–Ampère Keller–Segel
             equation},
   Journal = {Applied Mathematics Letters},
   Volume = {61},
   Pages = {26-34},
   Year = {2016},
   Month = {November},
   url = {http://dx.doi.org/10.1016/j.aml.2016.05.003},
   Doi = {10.1016/j.aml.2016.05.003},
   Key = {fds318453}
}

@article{fds323245,
   Author = {Huang, H and Liu, J-G},
   Title = {Error estimates of the aggregation-diffusion splitting
             algorithms for the Keller-Segel equations},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {21},
   Number = {10},
   Pages = {3463-3478},
   Year = {2016},
   Month = {November},
   url = {http://dx.doi.org/10.3934/dcdsb.2016107},
   Doi = {10.3934/dcdsb.2016107},
   Key = {fds323245}
}

@article{fds318454,
   Author = {Liu, J-G and Huang, H},
   Title = {Well-posedness for the Keller-Segel equation with fractional
             Laplacian and the theory of propagation of
             chaos},
   Journal = {Kinetic and Related Models},
   Volume = {9},
   Number = {4},
   Pages = {715-748},
   Year = {2016},
   Month = {September},
   url = {http://dx.doi.org/10.3934/krm.2016013},
   Doi = {10.3934/krm.2016013},
   Key = {fds318454}
}

@article{fds318455,
   Author = {Liu, J-G and Cong, W},
   Title = {A degenerate $p$-Laplacian Keller-Segel model},
   Journal = {Kinetic and Related Models},
   Volume = {9},
   Number = {4},
   Pages = {687-714},
   Year = {2016},
   Month = {September},
   url = {http://dx.doi.org/10.3934/krm.2016012},
   Doi = {10.3934/krm.2016012},
   Key = {fds318455}
}

@article{fds320551,
   Author = {Liu, J-G and Wang, J},
   Title = {A Note on L ∞ $L^{\infty}$ -Bound and Uniqueness to a
             Degenerate Keller-Segel Model},
   Journal = {Acta Applicandae Mathematicae},
   Volume = {142},
   Number = {1},
   Pages = {173-188},
   Year = {2016},
   Month = {April},
   ISSN = {0167-8019},
   url = {http://dx.doi.org/10.1007/s10440-015-0022-5},
   Doi = {10.1007/s10440-015-0022-5},
   Key = {fds320551}
}

@article{fds315797,
   Author = {Herschlag, G and Liu, J-G and Layton, AT},
   Title = {Fluid extraction across pumping and permeable walls in the
             viscous limit},
   Journal = {Physics of Fluids},
   Volume = {28},
   Number = {4},
   Pages = {041902-041902},
   Year = {2016},
   Month = {April},
   ISSN = {1070-6631},
   url = {http://dx.doi.org/10.1063/1.4946005},
   Doi = {10.1063/1.4946005},
   Key = {fds315797}
}

@article{fds320552,
   Author = {Liu, J-G and Pego, RL},
   Title = {On generating functions of Hausdorff moment
             sequences},
   Journal = {Transactions of the American Mathematical
             Society},
   Volume = {368},
   Number = {12},
   Pages = {8499-8518},
   Year = {2016},
   Month = {February},
   url = {http://dx.doi.org/10.1090/tran/6618},
   Doi = {10.1090/tran/6618},
   Key = {fds320552}
}

@article{fds329526,
   Author = {Chen, J and Liu, J-G and Zhou, Z},
   Title = {On a Schrödinger--Landau--Lifshitz System: Variational
             Structure and Numerical Methods},
   Journal = {Multiscale Modeling & Simulation},
   Volume = {14},
   Number = {4},
   Pages = {1463-1487},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M106947X},
   Doi = {10.1137/16M106947X},
   Key = {fds329526}
}

@article{fds323246,
   Author = {Liu, J-G and Xu, X},
   Title = {Existence Theorems for a Multidimensional Crystal Surface
             Model},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {48},
   Number = {6},
   Pages = {3667-3687},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1059400},
   Doi = {10.1137/16M1059400},
   Key = {fds323246}
}

@article{fds320553,
   Author = {Liu, JG and Zhang, Y},
   Title = {Convergence of diffusion-drift many particle systems in
             probability under a sobolev norm},
   Journal = {Springer Proceedings in Mathematics and Statistics},
   Volume = {162},
   Series = {Proceedings of Particle Systems and Partial Differential
             Equations - III},
   Pages = {195-223},
   Publisher = {Springer},
   Year = {2016},
   Month = {January},
   ISBN = {9783319321424},
   url = {http://dx.doi.org/10.1007/978-3-319-32144-8_10},
   Abstract = {© Springer International Publishing Switzerland 2016. In
             this paperwedevelop a newmartingale method to showthe
             convergence of the regularized empirical measure of many
             particle systems in probability under a Sobolev norm to the
             corresponding mean field PDE. Our method works well for the
             simple case of Fokker Planck equation and we can estimate a
             lower bound of the rate of convergence. This method can be
             generalized to more complicated systems with
             interactions.},
   Doi = {10.1007/978-3-319-32144-8_10},
   Key = {fds320553}
}

@article{fds320649,
   Author = {J.-G. Liu and R. Yang},
   Title = {Propagation of chaos for large Brownian particle system with
             Coulomb interaction},
   Journal = {Research in the Mathematical Sciences},
   Volume = {3},
   Number = {40},
   Year = {2016},
   Key = {fds320649}
}

@article{fds320549,
   Author = {Y. Duan and J.-G. Liu},
   Title = {Error estimate of the particle method for the
             b-equation},
   Journal = {Methods and Applications of Analysis},
   Volume = {23},
   Pages = {119-154},
   Year = {2016},
   Key = {fds320549}
}

@article{fds320556,
   Author = {J.-G. Liu and Y. Zhang},
   Title = {Convergence of stochastic interacting particle systems in
             probability under a Sobolev norm},
   Journal = {Annals of Mathematical Sciences and Applications},
   Volume = {1},
   Pages = {251-299},
   Year = {2016},
   Key = {fds320556}
}

@article{fds300225,
   Author = {J.-G. Liu and J. Wang},
   Title = {Refined hyper-contractivity and uniqueness for the
             Keller-Segel equations},
   Journal = {Applied Math Letter},
   Volume = {52},
   Pages = {212-219},
   Year = {2016},
   Key = {fds300225}
}

@article{fds246842,
   Author = {Xue, Y and Wang, C and Liu, J-G},
   Title = {Simple Finite Element Numerical Simulation of Incompressible
             Flow Over Non-rectangular Domains and the Super-Convergence
             Analysis},
   Journal = {Journal of Scientific Computing},
   Volume = {65},
   Number = {3},
   Pages = {1189-1216},
   Year = {2015},
   Month = {December},
   ISSN = {0885-7474},
   url = {http://dx.doi.org/10.1007/s10915-015-0005-8},
   Doi = {10.1007/s10915-015-0005-8},
   Key = {fds246842}
}

@article{fds246843,
   Author = {Lu, J and Liu, J-G and Margetis, D},
   Title = {Emergence of step flow from an atomistic scheme of epitaxial
             growth in 1+1 dimensions.},
   Journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter
             Physics},
   Volume = {91},
   Number = {3},
   Pages = {032403},
   Year = {2015},
   Month = {March},
   ISSN = {1539-3755},
   url = {http://dx.doi.org/10.1103/physreve.91.032403},
   Abstract = {The Burton-Cabrera-Frank (BCF) model for the flow of line
             defects (steps) on crystal surfaces has offered useful
             insights into nanostructure evolution. This model has rested
             on phenomenological grounds. Our goal is to show via scaling
             arguments the emergence of the BCF theory for noninteracting
             steps from a stochastic atomistic scheme of a kinetic
             restricted solid-on-solid model in one spatial dimension.
             Our main assumptions are: adsorbed atoms (adatoms) form a
             dilute system, and elastic effects of the crystal lattice
             are absent. The step edge is treated as a front that
             propagates via probabilistic rules for atom attachment and
             detachment at the step. We formally derive a quasistatic
             step flow description by averaging out the stochastic scheme
             when terrace diffusion, adatom desorption, and deposition
             from above are present.},
   Doi = {10.1103/physreve.91.032403},
   Key = {fds246843}
}

@article{fds300223,
   Author = {Degond, P and Frouvelle, A and Liu, JG},
   Title = {Phase Transitions, Hysteresis, and Hyperbolicity for
             Self-Organized Alignment Dynamics},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {216},
   Number = {1},
   Pages = {63-115},
   Year = {2015},
   Month = {January},
   ISSN = {0003-9527},
   url = {http://dx.doi.org/10.1007/s00205-014-0800-7},
   Abstract = {© 2014, Springer-Verlag Berlin Heidelberg. We provide a
             complete and rigorous description of phase transitions for
             kinetic models of self-propelled particles interacting
             through alignment. These models exhibit a competition
             between alignment and noise. Both the alignment frequency
             and noise intensity depend on a measure of the local
             alignment. We show that, in the spatially homogeneous case,
             the phase transition features (number and nature of
             equilibria, stability, convergence rate, phase diagram,
             hysteresis) are totally encoded in how the ratio between the
             alignment and noise intensities depend on the local
             alignment. In the spatially inhomogeneous case, we derive
             the macroscopic models associated to the stable equilibria
             and classify their hyperbolicity according to the same
             function.},
   Doi = {10.1007/s00205-014-0800-7},
   Key = {fds300223}
}

@article{fds300222,
   Author = {Chertock, A and Liu, JG and Pendleton, T},
   Title = {Elastic collisions among peakon solutions for the
             Camassa-Holm equation},
   Journal = {Applied Numerical Mathematics},
   Volume = {93},
   Pages = {30-46},
   Year = {2015},
   Month = {January},
   ISSN = {0168-9274},
   url = {http://dx.doi.org/10.1016/j.apnum.2014.01.001},
   Abstract = {© 2014 IMACS. The purpose of this paper is to study the
             dynamics of the interaction among a special class of
             solutions of the one-dimensional Camassa-Holm equation. The
             equation yields soliton solutions whose identity is
             preserved through nonlinear interactions. These solutions
             are characterized by a discontinuity at the peak in the wave
             shape and are thus called peakon solutions. We apply a
             particle method to the Camassa-Holm equation and show that
             the nonlinear interaction among the peakon solutions
             resembles an elastic collision, i.e., the total energy and
             momentum of the system before the peakon interaction is
             equal to the total energy and momentum of the system after
             the collision. From this result, we provide several
             numerical illustrations which support the analytical study,
             as well as showcase the merits of using a particle method to
             simulate solutions to the Camassa-Holm equation under a wide
             class of initial data.},
   Doi = {10.1016/j.apnum.2014.01.001},
   Key = {fds300222}
}

@article{fds313338,
   Author = {Herschlag, G and Liu, J-G and Layton, AT},
   Title = {An Exact Solution for Stokes Flow in a Channel with
             Arbitrarily Large Wall Permeability},
   Journal = {SIAM Journal on Applied Mathematics},
   Volume = {75},
   Number = {5},
   Pages = {2246-2267},
   Year = {2015},
   Month = {January},
   ISSN = {0036-1399},
   url = {http://dx.doi.org/10.1137/140995854},
   Doi = {10.1137/140995854},
   Key = {fds313338}
}

@article{fds246846,
   Author = {Degond, P and Liu, J-G and Ringhofer, C},
   Title = {Evolution of wealth in a non-conservative economy driven by
             local Nash equilibria},
   Journal = {Philosophical Transactions A},
   Volume = {372},
   Number = {2028},
   Pages = {20130394-20130394},
   Year = {2014},
   Month = {October},
   ISSN = {1364-503X},
   url = {http://dx.doi.org/10.1098/rsta.2013.0394},
   Doi = {10.1098/rsta.2013.0394},
   Key = {fds246846}
}

@article{fds246856,
   Author = {Goudon, T and Jin, S and Liu, JG and Yan, B},
   Title = {Asymptotic-preserving schemes for kinetic-fluid modeling of
             disperse two-phase flows with variable fluid
             density},
   Journal = {International Journal for Numerical Methods in
             Fluids},
   Volume = {75},
   Number = {2},
   Pages = {81-102},
   Year = {2014},
   Month = {May},
   ISSN = {0271-2091},
   url = {http://dx.doi.org/10.1002/fld.3885},
   Abstract = {We are concerned with a coupled system describing the
             interaction between suspended particles and a dense fluid.
             The particles are modeled by a kinetic equation of
             Vlasov-Fokker-Planck type, and the fluid is described by the
             incompressible Navier-Stokes system, with variable density.
             The systems are coupled through drag forces. High friction
             regimes lead to a purely hydrodynamic description of the
             mixture. We design first and second order
             asymptotic-preserving schemes suited to such regimes. We
             extend the method introduced in [Goudon T, Jin S, Liu JG,
             Yan B. Journal of Computational Physics 2013; 246:145-164]
             to the case of variable density in compressible flow. We
             check the accuracy and the asymptotic-preserving property
             numerically. We set up a few numerical experiments to
             demonstrate the ability of the scheme in capturing intricate
             interactions between the two phases on a wide range of
             physical parameters and geometric situations. © 2014 John
             Wiley & Sons, Ltd.},
   Doi = {10.1002/fld.3885},
   Key = {fds246856}
}

@article{fds246862,
   Author = {Duan, Y and Liu, JG},
   Title = {Convergence analysis of the vortex blob method for the
             b-equation},
   Journal = {Discrete and Continuous Dynamical Systems},
   Volume = {34},
   Number = {5},
   Pages = {1995-2011},
   Year = {2014},
   Month = {May},
   ISSN = {1078-0947},
   url = {http://dx.doi.org/10.3934/dcds.2014.34.1995},
   Abstract = {In this paper, we prove the convergence of the vortex blob
             method for a family of nonlinear evolutionary partial
             differential equations (PDEs), the so-called b-equation.
             This kind of PDEs, including the Camassa-Holm equation and
             the Degasperis-Procesi equation, has many applications in
             diverse scientific fields. Our convergence analysis also
             provides a proof for the existence of the global weak
             solution to the b-equation when the initial data is a
             nonnegative Radon measure with compact support.},
   Doi = {10.3934/dcds.2014.34.1995},
   Key = {fds246862}
}

@article{fds246848,
   Author = {Coquel, F and Jin, S and Liu, JG and Wang, L},
   Title = {Well-Posedness and Singular Limit of a Semilinear Hyperbolic
             Relaxation System with a Two-Scale Discontinuous Relaxation
             Rate},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {214},
   Number = {3},
   Pages = {1051-1084},
   Year = {2014},
   Month = {January},
   ISSN = {0003-9527},
   url = {http://dx.doi.org/10.1007/s00205-014-0773-6},
   Abstract = {© 2014, Springer-Verlag Berlin Heidelberg. Nonlinear
             hyperbolic systems with relaxations may encounter different
             scales of relaxation time, which is a prototype multiscale
             phenomenon that arises in many applications. In such a
             problem the relaxation time is of O(1) in part of the domain
             and very small in the remaining domain in which the solution
             can be approximated by the zero relaxation limit which can
             be solved numerically much more efficiently. For the
             Jin–Xin relaxation system in such a two-scale setting, we
             establish its wellposedness and singular limit as the
             (smaller) relaxation time goes to zero. The limit is a
             multiscale coupling problem which couples the original
             Jin–Xin system on the domain when the relaxation time is
             O(1) with its relaxation limit in the other domain through
             interface conditions which can be derived by matched
             interface layer analysis.As a result, we also establish the
             well-posedness and regularity (such as boundedness in sup
             norm with bounded total variation and L 1 -contraction) of
             the coupling problem, thus providing a rigorous mathematical
             foundation, in the general nonlinear setting, to the
             multiscale domain decomposition method for this two-scale
             problem originally proposed in Jin et al. in Math. Comp. 82,
             749–779, 2013.},
   Doi = {10.1007/s00205-014-0773-6},
   Key = {fds246848}
}

@article{fds246849,
   Author = {Degond, P and Herty, M and Liu, J-G},
   Title = {Flow on Sweeping Networks},
   Journal = {Multiscale Modeling & Simulation},
   Volume = {12},
   Number = {2},
   Pages = {538-565},
   Year = {2014},
   Month = {January},
   ISSN = {1540-3459},
   url = {http://dx.doi.org/10.1137/130927061},
   Doi = {10.1137/130927061},
   Key = {fds246849}
}

@article{fds225736,
   Author = {P. Degond and A. Frouvelle and J.-G. Liu},
   Title = {A note on phase transitions for the Smoluchowski equation
             with dipolar potential},
   Booktitle = {Proceedings of the Fourteenth International Conference on
             Hyperbolic Problems: Theory, Numerics and
             Application},
   Year = {2014},
   Key = {fds225736}
}

@article{fds246851,
   Author = {Chen, X and Li, X and Liu, J-G},
   Title = {Existence and uniqueness of global weak solution to a
             kinetic model for the sedimentation of rod-like
             particles},
   Journal = {Communications in Mathematical Sciences},
   Volume = {12},
   Number = {8},
   Pages = {1579-1601},
   Year = {2014},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/CMS.2014.v12.n8.a10},
   Doi = {10.4310/CMS.2014.v12.n8.a10},
   Key = {fds246851}
}

@article{fds246857,
   Author = {Johnston, H and Wang, C and Liu, J-G},
   Title = {A Local Pressure Boundary Condition Spectral Collocation
             Scheme for the Three-Dimensional Navier–Stokes
             Equations},
   Journal = {Journal of Scientific Computing},
   Volume = {60},
   Number = {3},
   Pages = {612-626},
   Year = {2014},
   ISSN = {0885-7474},
   url = {http://dx.doi.org/10.1007/s10915-013-9808-7},
   Abstract = {© 2014, Springer Science+Business Media New York.A spectral
             collocation scheme for the three-dimensional incompressible
             (u,p) formulation of the Navier–Stokes equations, in
             domains Ω with a non-periodic boundary condition, is
             described. The key feature is the high order approximation,
             by means of a local Hermite interpolant, of a Neumann
             boundary condition for use in the numerical solution of the
             pressure Poisson system. The time updates of the velocity u
             and pressure p are decoupled as a result of treating the
             pressure gradient in the momentum equation explicitly in
             time. The pressure update is computed from a pressure
             Poisson equation. Extension of the overall methodology to
             the Boussinesq system is also described. The uncoupling of
             the pressure and velocity time updates results in a highly
             efficient scheme that is simple to implement and well suited
             for simulating moderate to high Reynolds and Rayleigh number
             flows. Accuracy checks are presented, along with simulations
             of the lid-driven cavity flow and a differentially heated
             cavity flow, to demonstrate the scheme produces accurate
             three-dimensional results at a reasonable computational
             cost.},
   Doi = {10.1007/s10915-013-9808-7},
   Key = {fds246857}
}

@article{fds246858,
   Author = {Zou, C and Liu, J-G and Bian, S},
   Title = {Ultra-contractivity for Keller-Segel model with diffusion
             exponent $m>1-2/d$},
   Journal = {Kinetic and Related Models},
   Volume = {7},
   Number = {1},
   Pages = {9-28},
   Year = {2013},
   Month = {December},
   ISSN = {1937-5093},
   url = {http://dx.doi.org/10.3934/krm.2014.7.9},
   Doi = {10.3934/krm.2014.7.9},
   Key = {fds246858}
}

@article{fds246861,
   Author = {Huang, YL and Liu, JG and Wang, WC},
   Title = {A generalized mac scheme on curvilinear domains},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {35},
   Number = {5},
   Pages = {B953-B986},
   Year = {2013},
   Month = {November},
   ISSN = {1064-8275},
   url = {http://dx.doi.org/10.1137/120875508},
   Abstract = {We propose a simple finite difference scheme for
             Navier-Stokes equations in primitive formulation on
             curvilinear domains. With proper boundary treatment and
             interplay between covariant and contravariant components,
             the spatial discretization admits exact Hodge decomposition
             and energy identity. As a result, the pressure can be
             decoupled from the momentum equation with explicit time
             stepping. No artificial pressure boundary condition is
             needed. In addition, it can be shown that this spatially
             compatible discretization leads to uniform inf-sup
             condition, which plays a crucial role in the pressure
             approximation of both dynamic and steady state calculations.
             Numerical experiments demonstrate the robustness and
             efficiency of our scheme. Copyright © by SIAM. Unauthorized
             reproduction of this article is prohibited.},
   Doi = {10.1137/120875508},
   Key = {fds246861}
}

@article{fds220117,
   Author = {P. Degond and A. Frouvelle and J.-G. Liu and S Motsch and L
             Navoret},
   Title = {Macroscopic models of collective motion and
             self-organization},
   Journal = {Seminaire Laurent Schwartz -- EDP et applicatios},
   Volume = {2012 - 2013},
   Pages = {1-27},
   Year = {2013},
   Key = {fds220117}
}

@article{fds220119,
   Author = {P. Degond and J.-G, Liu and S. Motsch and V. Panferov},
   Title = {Hydrodynamic models of self-organized dynamics: derivation
             and existence theory},
   Journal = {Methods Anal. Appl.},
   Volume = {20},
   Pages = {89-114},
   Year = {2013},
   Key = {fds220119}
}

@article{fds246859,
   Author = {Degond, P and Liu, J-G and Ringhofer, C},
   Title = {Evolution of the Distribution of Wealth in an Economic
             Environment Driven by Local Nash Equilibria},
   Journal = {Journal of Statistical Physics},
   Volume = {154},
   Number = {3},
   Pages = {1-30},
   Year = {2013},
   ISSN = {0022-4715},
   url = {http://dx.doi.org/10.1007/s10955-013-0888-4},
   Abstract = {We present and analyze a model for the evolution of the
             wealth distribution within a heterogeneous economic
             environment. The model considers a system of rational agents
             interacting in a game theoretical framework, through fairly
             general assumptions on the cost function. This evolution
             drives the dynamic of the agents in both wealth and economic
             configuration variables. We consider a regime of scale
             separation where the large scale dynamics is given by a
             hydrodynamic closure with a Nash equilibrium serving as the
             local thermodynamic equilibrium. The result is a system of
             gas dynamics-type equations for the density and average
             wealth of the agents on large scales. We recover the inverse
             gamma distribution as an equilibrium in the particular case
             of quadratic cost functions which has been previously
             considered in the literature. © 2013 Springer
             Science+Business Media New York.},
   Doi = {10.1007/s10955-013-0888-4},
   Key = {fds246859}
}

@article{fds246860,
   Author = {Chen, X and Jüngel, A and Liu, J-G},
   Title = {A Note on Aubin-Lions-Dubinskiǐ Lemmas},
   Journal = {Acta Applicandae Mathematicae},
   Volume = {133},
   Number = {1},
   Pages = {1-11},
   Year = {2013},
   ISSN = {0167-8019},
   url = {http://dx.doi.org/10.1007/s10440-013-9858-8},
   Abstract = {Strong compactness results for families of functions in
             seminormed nonnegative cones in the spirit of the
             Aubin-Lions-Dubinskiǐ lemma are proven, refining some
             recent results in the literature. The first theorem sharpens
             slightly a result of Dubinskiǐ (in Mat. Sb.
             67(109):609-642, 1965) for seminormed cones. The second
             theorem applies to piecewise constant functions in time and
             sharpens slightly the results of Dreher and Jüngel (in
             Nonlinear Anal. 75:3072-3077, 2012) and Chen and Liu (in
             Appl. Math. Lett. 25:2252-2257, 2012). An application is
             given, which is useful in the study of porous-medium or
             fast-diffusion type equations. © 2013 Springer
             Science+Business Media.},
   Doi = {10.1007/s10440-013-9858-8},
   Key = {fds246860}
}

@article{fds246863,
   Author = {Degond, P and Liu, J-G and Ringhofer, C},
   Title = {Large-Scale Dynamics of Mean-Field Games Driven by Local
             Nash Equilibria},
   Journal = {Journal of Nonlinear Science},
   Volume = {24},
   Number = {1},
   Pages = {1-23},
   Year = {2013},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-013-9185-2},
   Abstract = {We introduce a new mean field kinetic model for systems of
             rational agents interacting in a game-theoretical framework.
             This model is inspired from non-cooperative anonymous games
             with a continuum of players and Mean-Field Games. The large
             time behavior of the system is given by a macroscopic
             closure with a Nash equilibrium serving as the local
             thermodynamic equilibrium. An application of the presented
             theory to a social model (herding behavior) is discussed. ©
             2013 Springer Science+Business Media New
             York.},
   Doi = {10.1007/s00332-013-9185-2},
   Key = {fds246863}
}

@article{fds246867,
   Author = {Chae, D and Degond, P and Liu, J-G},
   Title = {Well-posedness for Hall-magnetohydrodynamics},
   Journal = {Annales de l'Institut Henri Poincare (C) Analyse Non
             Lineaire},
   Volume = {31},
   Number = {3},
   Pages = {555-565},
   Year = {2013},
   ISSN = {0294-1449},
   url = {http://dx.doi.org/10.1016/j.anihpc.2013.04.006},
   Abstract = {We prove local existence of smooth solutions for large data
             and global smooth solutions for small data to the
             incompressible, resistive, viscous or inviscid Hall-MHD
             model. We also show a Liouville theorem for the stationary
             solutions. © 2013 Elsevier Masson SAS. All rights
             reserved.},
   Doi = {10.1016/j.anihpc.2013.04.006},
   Key = {fds246867}
}

@article{fds246864,
   Author = {Chen, X and Liu, J-G},
   Title = {Analysis of polymeric flow models and related compactness
             theorems in weighted spaces},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {45},
   Number = {3},
   Pages = {1179-1215},
   Year = {2013},
   ISSN = {0036-1410},
   url = {http://dx.doi.org/10.1137/120887850},
   Abstract = {We studied coupled systems of the Fokker-Planck equation and
             the Navier-Stokes equation modeling the Hookean and the
             finitely extensible nonlinear elastic (FENE)-type polymeric
             flows. We proved the continuous embedding and compact
             embedding theorems in weighted spaces that naturally arise
             from related entropy estimates. These embedding estimates
             are shown to be sharp. For the Hookean polymeric system with
             a center-of-mass diffusion and a superlinear spring
             potential, we proved the existence of a global weak
             solution. Moreover, we were able to tackle the FENE model
             with L2 initial data for the polymer density instead of the
             L∞ counterpart in the literature. © 2013 Society for
             Industrial and Applied Mathematics.},
   Doi = {10.1137/120887850},
   Key = {fds246864}
}

@article{fds246866,
   Author = {Bian, S and Liu, J-G},
   Title = {Dynamic and Steady States for Multi-Dimensional Keller-Segel
             Model with Diffusion Exponent m > 0},
   Journal = {Communications in Mathematical Physics},
   Volume = {323},
   Number = {3},
   Pages = {1017-1070},
   Year = {2013},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/s00220-013-1777-z},
   Abstract = {This paper investigates infinite-time spreading and
             finite-time blow-up for the Keller-Segel system. For 0 <
             m ≤ 2 - 2/d, the L p space for both dynamic and steady
             solutions are detected with (Formula presented.). Firstly,
             the global existence of the weak solution is proved for
             small initial data in L p. Moreover, when m > 1 - 2/d,
             the weak solution preserves mass and satisfies the
             hyper-contractive estimates in L q for any p < q <
             ∞. Furthermore, for slow diffusion 1 < m ≤ 2 - 2/d,
             this weak solution is also a weak entropy solution which
             blows up at finite time provided by the initial negative
             free energy. For m > 2 - 2/d, the hyper-contractive
             estimates are also obtained. Finally, we focus on the L p
             norm of the steady solutions, it is shown that the energy
             critical exponent m = 2d/(d + 2) is the critical exponent
             separating finite L p norm and infinite L p norm for the
             steady state solutions. © 2013 Springer-Verlag Berlin
             Heidelberg.},
   Doi = {10.1007/s00220-013-1777-z},
   Key = {fds246866}
}

@article{fds246869,
   Author = {Goudon, T and Jin, S and Liu, J-G and Yan, B},
   Title = {Asymptotic-preserving schemes for kinetic-fluid modeling of
             disperse two-phase flows},
   Journal = {Journal of Computational Physics},
   Volume = {246},
   Pages = {145-164},
   Year = {2013},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2013.03.038},
   Abstract = {We consider a system coupling the incompressible
             Navier-Stokes equations to the Vlasov-Fokker-Planck
             equation. Such a problem arises in the description of
             particulate flows. We design a numerical scheme to simulate
             the behavior of the system. This scheme is
             asymptotic-preserving, thus efficient in both the kinetic
             and hydrodynamic regimes. It has a numerical stability
             condition controlled by the non-stiff convection operator,
             with an implicit treatment of the stiff drag term and the
             Fokker-Planck operator. Yet, consistent to a standard
             asymptotic-preserving Fokker-Planck solver or an
             incompressible Navier-Stokes solver, only the
             conjugate-gradient method and fast Poisson and Helmholtz
             solvers are needed. Numerical experiments are presented to
             demonstrate the accuracy and asymptotic behavior of the
             scheme, with several interesting applications. © 2013
             Elsevier Inc.},
   Doi = {10.1016/j.jcp.2013.03.038},
   Key = {fds246869}
}

@article{fds246870,
   Author = {Chen, X and Liu, J-G},
   Title = {Global weak entropy solution to Doi-Saintillan-Shelley model
             for active and passive rod-like and ellipsoidal particle
             suspensions},
   Journal = {Journal of Differential Equations},
   Volume = {254},
   Number = {7},
   Pages = {2764-2802},
   Year = {2013},
   ISSN = {0022-0396},
   url = {http://dx.doi.org/10.1016/j.jde.2013.01.005},
   Abstract = {We prove the existence of the global weak entropy solution
             to the Doi-Saintillan-Shelley model for active and passive
             rod-like particle suspensions, which couples a Fokker-Planck
             equation with the incompressible Navier-Stokes or Stokes
             equation, under the no-flux boundary conditions,
             L2(Ω;L1(Sd-1)) initial data, and finite initial entropy for
             the particle distribution function in two and three
             dimensions. Furthermore, for the model with the Stokes
             equation, we obtain the global L2(Ω×Sd-1) weak solution in
             two and three dimensions and the uniqueness in two
             dimension. © 2013 Elsevier Inc..},
   Doi = {10.1016/j.jde.2013.01.005},
   Key = {fds246870}
}

@article{fds220112,
   Author = {A. Chertock and J.-G. Liu and T. Pendleton},
   Title = {Convergence analysis of the particle method for the
             Camassa-Holm equation},
   Pages = {365-373},
   Booktitle = {Proceedings of the 13th International Conference on
             ``Hyperbolic Problems: Theory, Numerics and
             Applications"},
   Publisher = {Higher Education Press},
   Address = {Beijing},
   Year = {2012},
   Key = {fds220112}
}

@article{fds246887,
   Author = {Chae, D and Liu, J-G},
   Title = {Blow-up, Zero α Limit and the Liouville Type Theorem for
             the Euler-Poincaré Equations},
   Journal = {Communications in Mathematical Physics},
   Volume = {314},
   Number = {3},
   Pages = {671-687},
   Year = {2012},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/s00220-012-1534-8},
   Abstract = {In this paper we study the Euler-Poincaré equations in ℝ
             N. We prove local existence of weak solutions in W 2,p(ℝ
             N),p>N, and local existence of unique classical solutions
             in H k(ℝ N),k> N/2+3, as well as a blow-up criterion.
             For the zero dispersion equation (α = 0) we prove a finite
             time blow-up of the classical solution. We also prove that
             as the dispersion parameter vanishes, the weak solution
             converges to a solution of the zero dispersion equation with
             sharp rate as α → 0, provided that the limiting solution
             belongs to C([0,T); H k(ℝ N)) with k > N/2 + 3. For the
             stationary weak solutions of the Euler-Poincaré equations
             we prove a Liouville type theorem. Namely, for α > 0 any
             weak solution u ∈ H 1(ℝ N) is u=0; for α= 0 any weak
             solution u ∈ L 2(ℝ N) is u=0. © 2012
             Springer-Verlag.},
   Doi = {10.1007/s00220-012-1534-8},
   Key = {fds246887}
}

@article{fds246888,
   Author = {Chen, X and Liu, J-G},
   Title = {Two nonlinear compactness theorems in L p(0,T;B)},
   Journal = {Applied Mathematics Letters},
   Volume = {25},
   Number = {12},
   Pages = {2252-2257},
   Year = {2012},
   ISSN = {0893-9659},
   url = {http://dx.doi.org/10.1016/j.aml.2012.06.012},
   Abstract = {We establish two nonlinear compactness theorems in L
             p(0,T;B) with hypothesis on time translations, which are
             nonlinear counterparts of two results by Simon (1987) [1].
             The first theorem sharpens a result by Maitre (2003) [10]
             and is important in the study of doubly nonlinear
             ellipticparabolic equations. Based on this theorem, we then
             obtain a time translation counterpart of a result by
             Dubinskiǐ (1965) [5], which is supposed to be useful in the
             study of some nonlinear kinetic equations (e.g. the
             FENE-type beadspring chains model). © 2012 Elsevier Ltd.
             All rights reserved.},
   Doi = {10.1016/j.aml.2012.06.012},
   Key = {fds246888}
}

@article{fds246889,
   Author = {Chen, L and Liu, J-G and Wang, J},
   Title = {Multidimensional degenerate Keller-Segel system with
             critical diffusion exponent 2n/(n + 2)},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {44},
   Number = {2},
   Pages = {1077-1102},
   Year = {2012},
   ISSN = {0036-1410},
   url = {http://dx.doi.org/10.1137/110839102},
   Abstract = {This paper deals with a degenerate diffusion
             Patlak-Keller-Segel system in n = 3 dimension. The main
             difference between the current work and many other recent
             studies on the same model is that we study the diffusion
             exponent m = 2n/(n + 2), which is smaller than the usual
             exponent m* = 2-2/n used in other studies. With the exponent
             m = 2n/(n + 2), the associated free energy is conformal
             invariant, and there is a family of stationary solutions
             Uλ,x0 (x) = C(λ/ λ 2+|x-x0| 2 ) n+2/2 λ < 0, σ0 ?
             ℝn. For radially symmetric solutions, we prove that if the
             initial data are strictly below Uλ,0(x) for some λ, then
             the solution vanishes in L1 loc as tλ8; if the initial data
             are strictly above Uλ,0(x) for some λ, then the solution
             either blows up at a finite time or has a mass concentration
             at r = 0 as time goes to infinity. For general initial data,
             we prove that there is a global weak solution provided that
             the Lm norm of initial density is less than a universal
             constant, and the weak solution vanishes as time goes to
             infinity. We also prove a finite time blow-up of the
             solution if the Lm norm for initial data is larger than the
             Lm norm of Uλ,x0 (x), which is constant independent of λ
             and x0, and the free energy of initial data is smaller than
             that of Uλ,x0(x). © 2012 Society for Industrial and
             Applied Mathematics.},
   Doi = {10.1137/110839102},
   Key = {fds246889}
}

@article{fds246890,
   Author = {Frouvelle, A and Liu, J-G},
   Title = {Dynamics in a kinetic model of oriented particles with phase
             transition},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {44},
   Number = {2},
   Pages = {791-826},
   Year = {2012},
   ISSN = {0036-1410},
   url = {http://dx.doi.org/10.1137/110823912},
   Abstract = {Motivated by a phenomenon of phase transition in a model of
             alignment of selfpropelled particles, we obtain a kinetic
             mean-field equation which is nothing more than the
             Smoluchowski equation on the sphere with dipolar potential.
             In this self-contained article, using only basic tools, we
             analyze the dynamics of this equation in any dimension. We
             first prove global wellposedness of this equation, starting
             with an initial condition in any Sobolev space. We then
             compute all possible steady states. There is a threshold for
             the noise parameter: over this threshold, the only
             equilibrium is the uniform distribution, and under this
             threshold, the other equilibria are the Fisher-von Mises
             distributions with arbitrary direction and a concentration
             parameter determined by the intensity of the noise. For any
             initial condition, we give a rigorous proof of convergence
             of the solution to a steady state as time goes to infinity.
             In particular, when the noise is under the threshold and
             with nonzero initial mean velocity, the solution converges
             exponentially fast to a unique Fisher- von Mises
             distribution. We also found a new conservation relation,
             which can be viewed as a convex quadratic entropy when the
             noise is above the threshold. This provides a uniform
             exponential rate of convergence to the uniform distribution.
             At the threshold, we show algebraic decay to the uniform
             distribution. © 2012 Society for Industrial and Applied
             Mathematics.},
   Doi = {10.1137/110823912},
   Key = {fds246890}
}

@article{fds246891,
   Author = {Carrillo, JA and Chen, L and Liu, J-G and Wang, J},
   Title = {A note on the subcritical two dimensional Keller-Segel
             system},
   Journal = {Acta Applicandae Mathematicae},
   Volume = {119},
   Number = {1},
   Pages = {43-55},
   Year = {2012},
   ISSN = {0167-8019},
   url = {http://dx.doi.org/10.1007/s10440-011-9660-4},
   Abstract = {The existence of solution for the 2D-Keller-Segel system in
             the subcritical case, i.e. when the initial mass is less
             than 8π, is reproved. Instead of using the entropy in the
             free energy and free energy dissipation, which was used in
             the proofs (Blanchet et al. in SIAM J. Numer. Anal.
             46:691-721, 2008; Electron. J. Differ. Equ. Conf. 44:32,
             2006 (electronic)), the potential energy term is fully
             utilized by adapting Delort's theory on 2D incompressible
             Euler equation (Delort in J. Am. Math. Soc. 4:553-386,
             1991). © 2011 Springer Science+Business Media
             B.V.},
   Doi = {10.1007/s10440-011-9660-4},
   Key = {fds246891}
}

@article{fds246892,
   Author = {Degond, P and Liu, J-G},
   Title = {Hydrodynamics of self-alignment interactions with precession
             and derivation of the Landau-Lifschitz-Gilbert
             equation},
   Journal = {Mathematical Models & Methods in Applied
             Sciences},
   Volume = {22},
   Number = {SUPPL.1},
   Pages = {1114001-18},
   Year = {2012},
   ISSN = {0218-2025},
   url = {http://dx.doi.org/10.1142/S021820251140001X},
   Abstract = {We consider a kinetic model of self-propelled particles with
             alignment interaction and with precession about the
             alignment direction. We derive a hydrodynamic system for the
             local density and velocity orientation of the particles. The
             system consists of the conservative equation for the local
             density and a non-conservative equation for the orientation.
             First, we assume that the alignment interaction is purely
             local and derive a first-order system. However, we show that
             this system may lose its hyperbolicity. Under the assumption
             of weakly nonlocal interaction, we derive diffusive
             corrections to the first-order system which lead to the
             combination of a heat flow of the harmonic map and
             LandauLifschitzGilbert dynamics. In the particular case of
             zero self-propelling speed, the resulting model reduces to
             the phenomenological LandauLifschitzGilbert equations.
             Therefore the present theory provides a kinetic formulation
             of classical micromagnetization models and spin dynamics. ©
             2012 World Scientific Publishing Company.},
   Doi = {10.1142/S021820251140001X},
   Key = {fds246892}
}

@article{fds246893,
   Author = {Chertock, A and Liu, J-G and Pendleton, T},
   Title = {Convergence of a particle method and global weak solutions
             of a family of evolutionary PDEs},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {50},
   Number = {1},
   Pages = {1-21},
   Year = {2012},
   ISSN = {0036-1429},
   url = {http://dx.doi.org/10.1137/110831386},
   Abstract = {The purpose of this paper is to provide global existence and
             uniqueness results for a family of fluid transport equations
             by establishing convergence results for the particle method
             applied to these equations. The considered family of PDEs is
             a collection of strongly nonlinear equations which yield
             traveling wave solutions and can be used to model a variety
             of flows in fluid dynamics. We apply a particle method to
             the studied evolutionary equations and provide a new
             self-contained method for proving its convergence. The
             latter is accomplished by using the concept of space-time
             bounded variation and the associated compactness properties.
             From this result, we prove the existence of a unique global
             weak solution in some special cases and obtain stronger
             regularity properties of the solution than previously
             established. © 2012 Society for Industrial and Applied
             Mathematics.},
   Doi = {10.1137/110831386},
   Key = {fds246893}
}

@article{fds246894,
   Author = {Haack, J and Jin, S and Liu, J-G},
   Title = {An all-speed asymptotic-preserving method for the isentropic
             Euler and Navier-Stokes equations},
   Journal = {Communications in computational physics},
   Volume = {12},
   Number = {4},
   Pages = {955-980},
   Year = {2012},
   ISSN = {1815-2406},
   url = {http://dx.doi.org/10.4208/cicp.250910.131011a},
   Abstract = {The computation of compressible flows becomes more
             challenging when the Mach number has different orders of
             magnitude. When the Mach number is of order one, modern
             shock capturing methods are able to capture shocks and other
             complex structures with high numerical resolutions. However,
             if the Mach number is small, the acoustic waves lead to
             stiffness in time and excessively large numerical viscosity,
             thus demanding much smaller time step and mesh size than
             normally needed for incompressible flow simulation. In this
             paper, we develop an all-speed asymptotic preserving (AP)
             numerical scheme for the compressible isentropic Euler and
             Navier-Stokes equations that is uniformly stable and
             accurate for all Mach numbers. Our idea is to split the
             system into two parts: one involves a slow, nonlinear and
             conservative hyperbolic system adequate for the use of
             modern shock capturing methods and the other a linear
             hyperbolic system which contains the stiff acoustic
             dynamics, to be solved implicitly. This implicit part is
             reformulated into a standard pressure Poisson projection
             system and thus possesses sufficient structure for efficient
             fast Fourier transform solution techniques. In the zero Mach
             number limit, the scheme automatically becomes a projection
             method-like incompressible solver. We present numerical
             results in one and two dimensions in both compressible and
             incompressible regimes. © 2012 Global-Science
             Press.},
   Doi = {10.4208/cicp.250910.131011a},
   Key = {fds246894}
}

@article{fds246895,
   Author = {Degond, P and Frouvelle, A and Liu, J-G},
   Title = {Macroscopic Limits and Phase Transition in a System of
             Self-propelled Particles},
   Journal = {Journal of Nonlinear Science},
   Volume = {23},
   Number = {3},
   Pages = {1-30},
   Year = {2012},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-012-9157-y},
   Abstract = {We investigate systems of self-propelled particles with
             alignment interaction. Compared to previous work (Degond and
             Motsch, Math. Models Methods Appl. Sci. 18:1193-1215, 2008a;
             Frouvelle, Math. Models Methods Appl. Sci., 2012), the force
             acting on the particles is not normalized, and this
             modification gives rise to phase transitions from disordered
             states at low density to aligned states at high densities.
             This model is the space-inhomogeneous extension of
             (Frouvelle and Liu, Dynamics in a kinetic model of oriented
             particles with phase transition, 2012), in which the
             existence and stability of the equilibrium states were
             investigated. When the density is lower than a threshold
             value, the dynamics is described by a nonlinear diffusion
             equation. By contrast, when the density is larger than this
             threshold value, the dynamics is described by a similar
             hydrodynamic model for self-alignment interactions as
             derived in (Degond and Motsch, Math. Models Methods Appl.
             Sci. 18:1193-1215, 2008a; Frouvelle, Math. Models Methods
             Appl. Sci., 2012). However, the modified normalization of
             the force gives rise to different convection speeds, and the
             resulting model may lose its hyperbolicity in some regions
             of the state space. © 2012 Springer Science+Business Media
             New York.},
   Doi = {10.1007/s00332-012-9157-y},
   Key = {fds246895}
}

@article{fds246885,
   Author = {Qin, HB and Wang, YG and Zhang, HB and Liu, J and Zhuo,
             Z},
   Title = {A passively mode-locked Nd:YVO 4 Laser with a
             carbon nanotube saturable absorber},
   Journal = {Laser Physics},
   Volume = {22},
   Number = {4},
   Pages = {684-687},
   Year = {2012},
   ISSN = {1054-660X},
   url = {http://dx.doi.org/10.1134/S1054660X12040160},
   Abstract = {We report the mode locking of a diode pumped Nd:YVO 4
             crystal laser by using a transmission-type single-walled
             carbon nanotube saturable absorber. The laser operated at
             1064 nm pumped by a fiber coupled laser diode with the
             cavity length of 1826 mm, generated a pulse width of 14 ps
             at a repetition rate of 82 MHz. The output power of 120 mW
             was obtained at the absorbed pumping power of 1400 mW. ©
             Pleiades Publishing, Ltd., 2012.},
   Doi = {10.1134/S1054660X12040160},
   Key = {fds246885}
}

@article{fds246871,
   Author = {Simsek, E and Liu, J and Liu, QH},
   Title = {A spectral integral method for the analysis of nano
             wires},
   Journal = {2011 30th URSI General Assembly and Scientific Symposium,
             URSIGASS 2011},
   Year = {2011},
   url = {http://dx.doi.org/10.1109/URSIGASS.2011.6050394},
   Abstract = {This work presents a spectrally accurate method for
             electromagnetic scattering from objects with complex
             permittivity embedded in a layered medium. Two-dimensional
             (2D) layered medium Green's functions are computed
             adaptively by using Gaussian quadratures. The singular terms
             in the Green's functions and the non-smooth terms in their
             derivatives are handled appropriately to achieve exponential
             convergence. Numerical results, compared with the ones
             obtained by using other methods, demonstrate the spectral
             accuracy and high efficiency of the proposed method. © 2011
             IEEE.},
   Doi = {10.1109/URSIGASS.2011.6050394},
   Key = {fds246871}
}

@article{fds246896,
   Author = {Jin, S and Liu, J-G and Wang, L},
   Title = {A domain decomposition method for semilinear hyperbolic
             systems with two-scale relaxations},
   Journal = {Math. Comp.},
   Volume = {82},
   Pages = {749-779},
   Year = {2011},
   Key = {fds246896}
}

@article{fds246897,
   Author = {Liu, J-G and Lorz, A},
   Title = {A coupled chemotaxis-fluid model: Global
             existence},
   Journal = {Annales de l'Institut Henri Poincare (C) Analyse Non
             Lineaire},
   Volume = {28},
   Number = {5},
   Pages = {643-652},
   Year = {2011},
   ISSN = {0294-1449},
   url = {http://dx.doi.org/10.1016/j.anihpc.2011.04.005},
   Abstract = {We consider a model arising from biology, consisting of
             chemotaxis equations coupled to viscous incompressible fluid
             equations through transport and external forcing. Global
             existence of solutions to the Cauchy problem is investigated
             under certain conditions. Precisely, for the
             chemotaxis-Navier- Stokes system in two space dimensions, we
             obtain global existence for large data. In three space
             dimensions, we prove global existence of weak solutions for
             the chemotaxis-Stokes system with nonlinear diffusion for
             the cell density.© 2011 Elsevier Masson SAS. All rights
             reserved.},
   Doi = {10.1016/j.anihpc.2011.04.005},
   Key = {fds246897}
}

@article{fds246898,
   Author = {Acheritogaray, M and Degond, P and Frouvelle, A and Liu,
             J-G},
   Title = {Kinetic formulation and global existence for the
             hall-magneto-hydrodynamics system},
   Journal = {Kinetic and Related Models},
   Volume = {4},
   Number = {4},
   Pages = {901-918},
   Year = {2011},
   ISSN = {1937-5093},
   url = {http://dx.doi.org/10.3934/krm.2011.4.901},
   Abstract = {This paper deals with the derivation and analysis of the the
             Hall Magneto-Hydrodynamic equations. We first provide a
             derivation of this system from a two-fluids Euler-Maxwell
             system for electrons and ions, through a set of scaling
             limits. We also propose a kinetic formulation for the
             Hall-MHD equa- tions which contains as fluid closure
             different variants of the Hall-MHD model. Then, we prove the
             existence of global weak solutions for the incompressible
             viscous resistive Hall-MHD model. We use the particular
             structure of the Hall term which has zero contribution to
             the energy identity. Finally, we discuss particular
             solutions in the form of axisymmetric purely swirling
             magnetic fields and propose some regularization of the Hall
             equation. © American Institute of Mathematical
             Sciences.},
   Doi = {10.3934/krm.2011.4.901},
   Key = {fds246898}
}

@article{fds246899,
   Author = {Zheng, W and Gao, H and Liu, J-G and Zhang, Y and Ye, Q and Swank,
             C},
   Title = {General solution to gradient-induced transverse and
             longitudinal relaxation of spins undergoing restricted
             diffusion},
   Journal = {Physical Review A - Atomic, Molecular, and Optical
             Physics},
   Volume = {84},
   Number = {5},
   Pages = {053411-8},
   Year = {2011},
   ISSN = {1050-2947},
   url = {http://dx.doi.org/10.1103/PhysRevA.84.053411},
   Abstract = {We develop an approach, by calculating the autocorrelation
             function of spins, to derive the magnetic field
             gradient-induced transverse (T2) relaxation of spins
             undergoing restricted diffusion. This approach is an
             extension to the method adopted by McGregor. McGregor's
             approach solves the problem only in the fast diffusion
             limit; however, our approach yields a single analytical
             solution suitable in all diffusion regimes, including the
             intermediate regime. This establishes a direct connection
             between the well-known slow diffusion result of Torrey and
             the fast diffusion result. We also perform free induction
             decay measurements on spin-exchange optically polarized 3He
             gas with different diffusion constants. The measured
             transverse relaxation profiles are compared with the theory
             and satisfactory agreement has been found throughout all
             diffusion regimes. In addition to the transverse relaxation,
             this approach is also applicable to solving the longitudinal
             relaxation (T 1) regardless of the diffusion limits. It
             turns out that the longitudinal relaxation in the slow
             diffusion limit differs by a factor of 2 from that in the
             fast diffusion limit. © 2011 American Physical
             Society.},
   Doi = {10.1103/PhysRevA.84.053411},
   Key = {fds246899}
}

@article{fds246904,
   Author = {Huang, Y-L and Liu, J-G and Wang, W-C},
   Title = {An FFT based fast poisson solver on spherical
             shells},
   Journal = {Communications in computational physics},
   Volume = {9},
   Number = {3},
   Pages = {649-667},
   Year = {2011},
   ISSN = {1815-2406},
   url = {http://dx.doi.org/10.4208/cicp.060509.080609s},
   Abstract = {We present a fast Poisson solver on spherical shells. With a
             special change of variable, the radial part of the Laplacian
             transforms to a constant coefficient differential operator.
             As a result, the Fast fourier Transform can be applied to
             solve the Poisson equation with O(N3log N) operations.
             Numerical examples have confirmed the accuracy and
             robustness of the new scheme. © 2011 Global-Science
             Press.},
   Doi = {10.4208/cicp.060509.080609s},
   Key = {fds246904}
}

@article{fds246900,
   Author = {Liu, J-G and Liu, J and Pego, RL},
   Title = {Stable and accurate pressure approximation for unsteady
             incompressible viscous flow},
   Journal = {Journal of Computational Physics},
   Volume = {229},
   Number = {9},
   Pages = {3428-3453},
   Year = {2010},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2010.01.010},
   Abstract = {How to properly specify boundary conditions for pressure is
             a longstanding problem for the incompressible Navier-Stokes
             equations with no-slip boundary conditions. An analytical
             resolution of this issue stems from a recently developed
             formula for the pressure in terms of the commutator of the
             Laplacian and Leray projection operators. Here we make use
             of this formula to (a) improve the accuracy of computing
             pressure in two kinds of existing time-discrete projection
             methods implicit in viscosity only, and (b) devise new
             higher-order accurate time-discrete projection methods that
             extend a slip-correction idea behind the well-known
             finite-difference scheme of Kim and Moin. We test these
             schemes for stability and accuracy using various
             combinations of C0 finite elements. For all three kinds of
             time discretization, one can obtain third-order accuracy for
             both pressure and velocity without a time-step stability
             restriction of diffusive type. Furthermore, two kinds of
             projection methods are found stable using piecewise-linear
             elements for both velocity and pressure. © 2010 Elsevier
             Inc.},
   Doi = {10.1016/j.jcp.2010.01.010},
   Key = {fds246900}
}

@article{fds246905,
   Author = {Liu, J-G and Mieussens, L},
   Title = {Analysis of an asymptotic preserving scheme for linear
             kinetic equations in the diffusion limit},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {48},
   Number = {4},
   Pages = {1474-1491},
   Year = {2010},
   ISSN = {0036-1429},
   url = {http://hdl.handle.net/10161/4316 Duke open
             access},
   Abstract = {We present a mathematical analysis of the asymptotic
             preserving scheme proposed in [M. Lemou and L. Mieussens,
             SIAM J. Sci. Comput., 31 (2008), pp. 334-368] for linear
             transport equations in kinetic and diffusive regimes. We
             prove that the scheme is uniformly stable and accurate with
             respect to the mean free path of the particles. This
             property is satisfied under an explicitly given CFL
             condition. This condition tends to a parabolic CFL condition
             for small mean free paths and is close to a convection CFL
             condition for large mean free paths. Our analysis is based
             on very simple energy estimates. © 2010 Society for
             Industrial and Applied Mathematics.},
   Doi = {10.1137/090772770},
   Key = {fds246905}
}

@article{fds246928,
   Author = {Liu, J-G and Pego, R},
   Title = {Stable discretization of magnetohydrodynamics in bounded
             domains},
   Journal = {Commun. Math. Sci.},
   Volume = {8},
   Number = {1},
   Pages = {234-251},
   Year = {2010},
   ISSN = {1539-6746},
   Abstract = {We study a semi-implicit time-difference scheme for
             magnetohydrodynamics of a viscous and resistive
             incompressible fluid in a bounded smooth domain with a
             perfectly conducting boundary. In the scheme, the velocity
             and magnetic fields are updated by solving simple Helmholtz
             equations. Pressure is treated explicitly in time, by
             solving Poisson equations corresponding to a recently
             de-veloped formula for the Navier-Stokes pressure involving
             the commutator of Laplacian and Leray projection operators.
             We prove stability of the time-difference scheme, and deduce
             a local-time well-posedness theorem for MHD dynamics
             extended to ignore the divergence-free constraint on
             velocity and magnetic fields. These fields are
             divergence-free for all later time if they are initially so.
             © 2010 International Press.},
   Key = {fds246928}
}

@article{fds304584,
   Author = {Liu, J-G and Pego, RL},
   Title = {Stable discretization of magnetohydrodynamics in bounded
             domains},
   Journal = {Communications in Mathematical Sciences},
   Volume = {8},
   Number = {1},
   Pages = {235-251},
   Year = {2010},
   ISSN = {1539-6746},
   Abstract = {We study a semi-implicit time-difference scheme for
             magnetohydrodynamics of a viscous and resistive
             incompressible fluid in a bounded smooth domain with a
             perfectly conducting boundary. In the scheme, the velocity
             and magnetic fields are updated by solving simple Helmholtz
             equations. Pressure is treated explicitly in time, by
             solving Poisson equations corresponding to a recently
             de-veloped formula for the Navier-Stokes pressure involving
             the commutator of Laplacian and Leray projection operators.
             We prove stability of the time-difference scheme, and deduce
             a local-time well-posedness theorem for MHD dynamics
             extended to ignore the divergence-free constraint on
             velocity and magnetic fields. These fields are
             divergence-free for all later time if they are initially so.
             © 2010 International Press.},
   Key = {fds304584}
}

@article{fds246882,
   Author = {Liu, QH and Lin, Y and Liu, J and Lee, J-H and Şimşek,
             E},
   Title = {A 3-D spectral integral method (SIM) for surface integral
             equations},
   Journal = {IEEE Microwave and Wireless Components Letters},
   Volume = {19},
   Number = {2},
   Pages = {62-64},
   Year = {2009},
   ISSN = {1531-1309},
   url = {http://dx.doi.org/10.1109/LMWC.2008.2011305},
   Abstract = {An efficient 3-D spectral integral method (SIM) has been
             proposed to speed up the method of moments (MOM) calculation
             of induced currents on a cuboid. This method utilizes the
             Toeplitz structure in the impedance matrix and the fast
             Fourier transform to accelerate the MOM solution. It reduces
             the memory and CPU time per iteration from O(N2) in the MOM
             to O(N1.5) and O(N1.5 log N), respectively. Thus, the SIM
             can be also used as an efficient radiation boundary
             condition for the finite element method. Numerical results
             confirm the effectiveness of this method. © 2006
             IEEE.},
   Doi = {10.1109/LMWC.2008.2011305},
   Key = {fds246882}
}

@article{fds246883,
   Author = {Lin, Y and Lee, J-H and Liu, J and Chai, M and Mix, JA and Liu,
             QH},
   Title = {A hybrid SIM-SEM method for 3-D electromagnetic scattering
             problems},
   Journal = {IEEE Transactions on Antennas and Propagation},
   Volume = {57},
   Number = {11},
   Pages = {3655-3663},
   Year = {2009},
   ISSN = {0018-926X},
   url = {http://dx.doi.org/10.1109/TAP.2009.2026664},
   Abstract = {A new method combining the spectral integral method and
             spectral element method (SIM-SEM) is proposed to simulate
             3-D electromagnetic scattering from inhomogeneous objects.
             In this hybrid technique (a special case of the finite
             element boundary integral (FEM-BI) combination), the SEM
             with the mixed-order curl conforming vector
             Gauss-Lobatto-Legendre (GLL) basis functions are used to
             represent the interior electric field with high accuracy,
             while the SIM on a cuboid surface is used as an exact
             radiation boundary condition. The Toeplitz property of the
             SIM matrix is utilized to reduce the memory and CPU time
             costs in an iterative solver by using the fast Fourier
             transform algorithm. Unlike the traditional FEM-BI
             combination where the BI portion usually dominates the
             computational complexity, the computational costs are much
             lower in the SIM-SEM method. Numerical results verify the
             accuracy and capability of this method, confirming that the
             SIM-SEM method is a good alternative for solving scattering
             problems from inhomogeneous objects. © 2006
             IEEE.},
   Doi = {10.1109/TAP.2009.2026664},
   Key = {fds246883}
}

@article{fds246943,
   Author = {Liu, J-G and Liu, J and Pego, RL},
   Title = {Error estimates for finite-element Navier-Stokes solvers
             without standard Inf-Sup conditions},
   Journal = {Chinese Annals of Mathematics - Series B},
   Volume = {30},
   Number = {6},
   Pages = {743-768},
   Year = {2009},
   ISSN = {0252-9599},
   url = {http://dx.doi.org/10.1007/s11401-009-0116-3},
   Abstract = {The authors establish error estimates for recently developed
             finite-element methods for incompressible viscous flow in
             domains with no-slip boundary conditions. The methods arise
             by discretization of a well-posed extended Navier-Stokes
             dynamics for which pressure is determined from current
             velocity and force fields. The methods use C1 elements for
             velocity and C0 elements for pressure. A stability estimate
             is proved for a related finite-element projection method
             close to classical time-splitting methods of Orszag,
             Israeli, DeVille and Karniadakis. © Editorial Office of CAM
             and Springer-Verlag Berlin Heidelberg 2009.},
   Doi = {10.1007/s11401-009-0116-3},
   Key = {fds246943}
}

@article{fds246944,
   Author = {Liu, J-G and Wang, W-C},
   Title = {Characterization and regularity for axisymmetric solenoidal
             vector fields with application to navier-stokes
             equation},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {41},
   Number = {5},
   Pages = {1825-1850},
   Year = {2009},
   ISSN = {0036-1410},
   url = {http://dx.doi.org/10.1137/080739744},
   Abstract = {We consider the vorticity-stream formulation of axisymmetric
             incompressible flows and its equivalence with the primitive
             formulation. It is shown that, to characterize the
             regularity of a divergence free axisymmetric vector field in
             terms of the swirling components, an extra set of pole
             conditions is necessary to give a full description of the
             regu larity. In addition, smooth solutions up to the axis of
             rotation give rise to smooth solutions of primitive
             formulation in the case of the Navier-Stokes equation, but
             not the Euler equation. We also establish a proper weak
             formulation and show its equivalence to Leray's formulation.
             © 2009 Society for Industrial and Applied
             Mathematics.},
   Doi = {10.1137/080739744},
   Key = {fds246944}
}

@article{fds246945,
   Author = {Ha, S-Y and Liu, J-G},
   Title = {A simple proof of the Cucker-Smale flocking dynamics and
             mean-field limit},
   Journal = {Communications in Mathematical Sciences},
   Volume = {7},
   Number = {2},
   Pages = {297-325},
   Year = {2009},
   ISSN = {1539-6746},
   Abstract = {We present a simple proof on the formation of flocking to
             the Cucker-Smale system based on the explicit construction
             of a Lyapunov functional. Our results also provide a unified
             condition on the initial states in which the exponential
             convergence to flocking state will occur. For large particle
             systems, we give a rigorous justification for the mean-field
             limit from the many particle Cucker-Smale system to the
             Vlasov equation with flocking dissipation as the number of
             particles goes to infinity. © 2009 International
             Press.},
   Key = {fds246945}
}

@article{fds246940,
   Author = {Hsia, C-H and Liu, J-G and Wang, C},
   Title = {Structural stability and bifurcation for 2D incompressible
             ows with symmetry},
   Journal = {Meth. Appl. Anal.},
   Volume = {15},
   Pages = {495-512},
   Year = {2008},
   Key = {fds246940}
}

@article{fds246941,
   Author = {Lin, P and Liu, J-G and Lu, X},
   Title = {Long time numerical solution of the Navier-Stokes equations
             based on a sequential regularization formulation},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {31},
   Number = {1},
   Pages = {398-419},
   Year = {2008},
   ISSN = {1064-8275},
   url = {http://dx.doi.org/10.1137/060673722},
   Abstract = {The sequential regularization method is a reformulation of
             the unsteady Navier-Stokes equations from the viewpoint of
             constrained dynamical systems or the approximate
             Helmholtz-Hodge projection. In this paper we study the long
             time behavior of the sequential regularization formulation.
             We give a uniform-in-time estimate between the solution of
             the reformulated system and that of the Navier-Stokes
             equations. We also conduct an error analysis for the
             temporal discrete system and show that the error bound is
             independent of time. A couple of long time flow examples are
             computed to demonstrate this method. © 2008 Society for
             Industrial and Applied Mathematics.},
   Doi = {10.1137/060673722},
   Key = {fds246941}
}

@article{fds246942,
   Author = {Liu, J-G and Wang, C},
   Title = {A fourth order numerical method for the primtive equations
             formulated in mean vorticity},
   Journal = {Communications in computational physics},
   Volume = {4},
   Number = {1},
   Pages = {26-55},
   Year = {2008},
   ISSN = {1815-2406},
   Abstract = {A fourth-order finite difference method is proposed and
             studied for the primitive equations (PEs) of large-scale
             atmospheric and oceanic flow based on mean vorticity
             formulation. Since the vertical average of the horizontal
             velocity field is divergence-free, we can introduce mean
             vorticity and mean stream function which are connected by a
             2-D Poisson equation. As a result, the PEs can be
             reformulated such that the prognostic equation for the
             horizontal velocity is replaced by evolutionary equations
             for the mean vorticity field and the vertical derivative of
             the horizontal velocity. The mean vorticity equation is
             approximated by a compact difference scheme due to the
             difficulty of the mean vorticity boundary condition, while
             fourth-order long-stencil approximations are utilized to
             deal with transport type equations for computational
             convenience. The numerical values for the total velocity
             field (both horizontal and vertical) are statically
             determined by a discrete realization of a differential
             equation at each fixed horizontal point. The method is
             highly efficient and is capable of producing highly resolved
             solutions at a reasonable computational cost. The full
             fourth-order accuracy is checked by an example of the
             reformulated PEs with force terms. Additionally, numerical
             results of a large-scale oceanic circulation are presented.
             © 2008 Global-Science Press.},
   Key = {fds246942}
}

@article{fds246946,
   Author = {Degond, P and Liu, J-G and Vignal, M-H},
   Title = {Analysis of an asymptotic preserving scheme for the
             Euler-Poisson system in the quasineutral
             limit},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {46},
   Number = {3},
   Pages = {1298-1322},
   Year = {2008},
   ISSN = {0036-1429},
   url = {http://dx.doi.org/10.1137/070690584},
   Keywords = {stiffness • Debye length • electron plasma period
             • Burgers-Poisson • sheath problem •
             Klein-Gordon},
   Abstract = {In a previous work [P. Crispel, P. Degond, and M.-H. Vignal,
             J. Comput. Phys., 223 (2007), pp. 208-234], a new numerical
             discretization of the Euler-Poisson system was proposed.
             This scheme is "asymptotic preserving" in the quasineutral
             limit (i.e., when the Debye length ε tends to zero), which
             means that it becomes consistent with the limit model when
             ε → 0. In the present work, we show that the stability
             domain of the present scheme is independent of ε. This
             stability analysis is performed on the Fourier transformed
             (with respect to the space variable) linearized system. We
             show that the stability property is more robust when a
             space-decentered scheme is used (which brings in some
             numerical dissipation) rather than a space-centered scheme.
             The linearization is first performed about a zero mean
             velocity and then about a nonzero mean velocity. At the
             various stages of the analysis, our scheme is compared with
             more classical schemes and its improved stability property
             is outlined. The analysis of a fully discrete (in space and
             time) version of the scheme is also given. Finally, some
             considerations about a model nonlinear problem, the
             Burgers-Poisson problem, are also discussed. © 2008 Society
             for Industrial and Applied Mathematics.},
   Doi = {10.1137/070690584},
   Key = {fds246946}
}

@article{fds246948,
   Author = {Lu, X and Lin, P and Liu, J-G},
   Title = {Analysis of a sequential regularization method for the
             unsteady Navier-Stokes equations},
   Journal = {Mathematics of Computation},
   Volume = {77},
   Number = {263},
   Pages = {1467-1494},
   Year = {2008},
   ISSN = {0025-5718},
   url = {http://dx.doi.org/10.1090/S0025-5718-08-02087-5},
   Keywords = {Navier-Stokes equations • iterative penalty method
             • implicit parabolic PDE • error estimates •
             constrained dynamical system • stabilization
             method},
   Abstract = {The incompressibility constraint makes Navier-Stokes
             equations difficult. A reformulation to a better posed
             problem is needed before solving it numerically. The
             sequential regularization method (SRM) is a reformulation
             which combines the penalty method with a stabilization
             method in the context of constrained dynamical systems and
             has the benefit of both methods. In the paper, we study the
             existence and uniqueness for the solution of the SRM and
             provide a simple proof of the convergence of the solution of
             the SRM to the solution of the Navier-Stokes equations. We
             also give error estimates for the time discretized SRM
             formulation. ©2008 American Mathematical
             Society.},
   Doi = {10.1090/S0025-5718-08-02087-5},
   Key = {fds246948}
}

@article{fds139011,
   Author = {J.-G. Liu and Jie Liu and R. Pego},
   Title = {Estimates on the Stokes pressure by partitioning the energy
             of harmonic functions},
   Pages = {251--270},
   Booktitle = {Kyoto Conference on the Navier-Stokes equations and their
             Applications},
   Publisher = {Kyoto Univ.},
   Editor = {Y. Giga and H. Kozono and H. Okamoto and Y. Shibta},
   Year = {2007},
   Abstract = {We show that in a tubular domain with sufficiently small
             width, the normal and tangential gradients of a harmonic
             function have almost the same L2 norm. This estimate yields
             a sharp estimate of the pressure in terms of the viscosity
             term in the Navier-Stokes equation with no-slip boundary
             condition. By consequence, one can analyze the Navier-
             Stokes equations simply as a perturbed vector diffusion
             equation instead of as a perturbed Stokes system. As an
             application, we describe a rather easy approach to establish
             a new isomorphism theorem for the non-homogeneous Stokes
             system.},
   Key = {fds139011}
}

@article{fds246878,
   Author = {Liu, J and Lin, Y and Lee, J-H and Simsek, E and Liu,
             QH},
   Title = {Application of the hybrid spectral integral method with
             spectral element method},
   Journal = {International Symposium on Antennas and Propagation
             (APSURSI)},
   Pages = {5611-},
   Year = {2007},
   ISSN = {1522-3965},
   url = {http://dx.doi.org/10.1109/APS.2007.4396821},
   Doi = {10.1109/APS.2007.4396821},
   Key = {fds246878}
}

@article{fds246880,
   Author = {Liu, J-G and Liu, J and Pego, RL},
   Title = {Stability and convergence of efficient Navier-Stokes solvers
             via a commutator estimate},
   Journal = {Communications on Pure & Applied Mathematics},
   Volume = {60},
   Number = {10},
   Pages = {1443-1487},
   Year = {2007},
   ISSN = {0010-3640},
   url = {http://dx.doi.org/10.1002/cpa.20178},
   Abstract = {For strong solutions of the incompressible Navier-Stokes
             equations in bounded domains with velocity specified at the
             boundary, we establish the unconditional stability and
             convergence of discretization schemes that decouple the
             updates of pressure and velocity through explicit time
             stepping for pressure. These schemes require no solution of
             stationary Stokes systems, nor any compatibility between
             velocity and pressure spaces to ensure an inf-sup condition,
             and are representative of a class of highly efficient
             computational methods that have recently emerged. The proofs
             are simple, based upon a new, sharp estimate for the
             commutator of the Laplacian and Helmholtz projection
             operators. This allows us to treat an unconstrained
             formulation of the Navier-Stokes equations as a perturbed
             diffusion equation. ©2006 Wiley Periodicals,
             Inc.},
   Doi = {10.1002/cpa.20178},
   Key = {fds246880}
}

@article{fds246881,
   Author = {Liu, J and Liu, QH},
   Title = {A novel radiation boundary condition for finite-element
             method},
   Journal = {Microwave and Optical Technology Letters},
   Volume = {49},
   Number = {8},
   Pages = {1995-2002},
   Year = {2007},
   ISSN = {0895-2477},
   url = {http://dx.doi.org/10.1002/mop.22608},
   Abstract = {This paper presents a novel radiation boundary condition
             (RBC) with the spectral integral method (SIM) to truncate
             the computational domain in the finite-element method (FEM).
             Because of the spectral accuracy of the SIM, the sampling
             density on the radiation boundary requires less than four
             points per wavelength to achieve a high accuracy (1%). As a
             result, the introduction of the SIM as an RBC actually can
             decrease the total number of unknowns in the system
             equation. Numerical results illustrate the usefulness of
             this novel RBC for both homogeneous and inhomogeneous
             objects. © 2007 Wiley Periodicals, Inc.},
   Doi = {10.1002/mop.22608},
   Key = {fds246881}
}

@article{fds246903,
   Author = {Liu, J-G and Liu, J and Pego, R},
   Title = {Stability and convergence of efficient Navier-Stokes solvers
             via a commutator estimate via a commutator
             estimate},
   Journal = {Comm. Pure Appl. Math.},
   Volume = {60},
   Pages = {1443-1487},
   Year = {2007},
   Key = {fds246903}
}

@article{fds246947,
   Author = {Degond, P and Jin, S and Liu, JG},
   Title = {Mach-number uniform asymptotic- preserving Gauge schemes for
             compressible flows},
   Journal = {Bulletin of the Institute of Mathematics Academia Sinica
             (New Series)},
   Volume = {2},
   Pages = {851-892},
   Year = {2007},
   Keywords = {Mach number uniform method • Euler equations •
             Navier-Stokes equations • Asymptotic Preserving schemes
             • gauge schemes • compressible fluids •
             Low-Mach number limit • macro-micro decomposition
             • semi-implicit scheme • Euler-Poisson system
             • Navier-Stokes-Poisson system},
   Abstract = {We present novel algorithms for compressible flows that are
             efficient for all Mach numbers. The approach is based on
             several ingredients: semi-implicit schemes, the gauge
             decomposition of the velocity field and a second order
             formulation of the density equation (in the isentropic case)
             and of the energy equation (in the full Navier-Stokes case).
             Additionally, we show that our approach corresponds to a
             micro-macro decomposition of the model, where the macro
             field corresponds to the incompressible component satisfying
             a perturbed low Mach number limit equation and the micro
             field is the potential component of the velocity. Finally,
             we also use the conservative variables in order to obtain a
             proper conservative formulation of the equations when the
             Mach number is order unity. We successively consider the
             isentropic case, the full Navier-Stokes case, and the
             isentropic Navier-Stokes-Poisson case. In this work, we only
             concentrate on the question of the time discretization and
             show that the proposed method leads to Asymptotic Preserving
             schemes for compressible flows in the low Mach number
             limit.},
   Key = {fds246947}
}

@article{fds246949,
   Author = {Antman, SS and Liu, J-G},
   Title = {Basic themes and pretty problems of nonlinear solid
             mechanics},
   Journal = {Milan Journal of Mathematics},
   Volume = {75},
   Number = {1},
   Pages = {135-176},
   Year = {2007},
   ISSN = {1424-9286},
   url = {http://dx.doi.org/10.1007/s00032-007-0068-6},
   Keywords = {Nonlinear solid mechanics • radial motions •
             existence • multiplicity • blowup • inverse
             problems • quasistaticity • control •
             invariant artificial viscosity and shock
             structure},
   Abstract = {The first part of this paper describes some important
             underlying themes in the mathematical theory of continuum
             mechanics that are distinct from formulating and analyzing
             governing equations. The main part of this paper is devoted
             to a survey of some concrete, conceptually simple, pretty
             problems that help illuminate the underlying themes. The
             paper concludes with a discussion of the crucial role of
             invariant constitutive equations in computation. © 2007
             Birkhaueser.},
   Doi = {10.1007/s00032-007-0068-6},
   Key = {fds246949}
}

@article{fds246958,
   Author = {Moore, J and Cheng, Z and Hao, J and Guo, G and Liu, J-G and Lin, C and Yu,
             L},
   Title = {Effects of solid-state yeast treatment on the antioxidant
             properties and protein and fiber compositions of common hard
             wheat bran},
   Journal = {Journal of Agricultural and Food Chemistry},
   Volume = {55},
   Number = {25},
   Pages = {10173-10182},
   Year = {2007},
   ISSN = {0021-8561},
   url = {http://dx.doi.org/10.1021/jf071590o},
   Abstract = {The bran fraction of wheat grain is known to contain
             significant quantities of bioactive components. This study
             evaluated the potential of solid-state yeast fermentation to
             improve the health beneficial properties of wheat bran,
             including extractable antioxidant properties, protein
             contents, and soluble and insoluble fiber compositions.
             Three commercial food grade yeast preparations were
             evaluated in the study along with the effects of yeast dose,
             treatment time, and their interaction with the beneficial
             components. Solid-state yeast treatments were able to
             significantly increase releasable antioxidant properties
             ranging from 28 to 65, from 0 to 20, from 13 to 19, from 0
             to 25, from 50 to 100, and from 3 to 333% for scavenging
             capacities against peroxyl (ORAC), ABTS cation, DPPH and
             hydroxyl radicals, total phenolic contents (TPC), and
             phenolic acids, respectively. Yeast treatment increased
             protein content 11-12% but did not significantly alter the
             fiber composition of wheat bran. Effects of solid-state
             yeast treatment on both ORAC and TPC of wheat bran were
             altered by yeast dose, treatment time, and their
             interaction. Results suggest that solid-state yeast
             treatment may be a commercially viable postharvest procedure
             for improving the health beneficial properties of wheat bran
             and other wheat-based food ingredients. © 2007 American
             Chemical Society.},
   Doi = {10.1021/jf071590o},
   Key = {fds246958}
}

@article{fds139013,
   Author = {J.-G. Liu and Jie Liu and R. Pego},
   Title = {On incompressible Navier-Stokes dynamics: a new approach for
             analysis and computation},
   Pages = {29--44},
   Booktitle = {Proceedings of the Tenth International Conference on
             Hyperbolic Problems},
   Publisher = {Yokohama Publishers, Inc.},
   Editor = {F. Asakura and etc},
   Year = {2006},
   Key = {fds139013}
}

@article{fds246879,
   Author = {Şimşek, E and Liu, J and Liu, QH},
   Title = {A spectral integral method and hybrid SIM/FEM for layered
             media},
   Journal = {IEEE Transactions on Microwave Theory and
             Techniques},
   Volume = {54},
   Number = {11},
   Pages = {3878-3884},
   Year = {2006},
   ISSN = {0018-9480},
   url = {http://dx.doi.org/10.1109/TMTT.2006.883647},
   Abstract = {This paper first presents a spectral integral method (SIM)
             for electromagnetic scattering from homogeneous dielectric
             and perfectly electric conducting objects straddling several
             layers of a multilayered medium. It then uses this SIM as an
             exact radiation boundary condition to truncate the
             computational domain in the finite-element method (FEM) to
             form a hybrid SIM/FEM, which is applicable to arbitrary
             inhomogeneous objects. Due to the high accuracy of the SIM,
             the sampling density on the radiation boundary requires less
             than five points per wavelength to achieve 1% accuracy. The
             efficiency and accuracy of the developed methods have been
             demonstrated with several numerical experiments for the TM Z
             case. The TE Z case can be obtained by duality. © 2006
             IEEE.},
   Doi = {10.1109/TMTT.2006.883647},
   Key = {fds246879}
}

@article{fds246901,
   Author = {Degond, P and Liu, J-G and Mieussens, L},
   Title = {Macroscopic fluid models with localized kinetic upscaling
             effects},
   Journal = {Multiscale Modeling & Simulation},
   Volume = {5},
   Number = {3},
   Pages = {940-979},
   Year = {2006},
   ISSN = {1540-3459},
   url = {http://dx.doi.org/10.1137/060651574},
   Keywords = {Kinetic-Fluid coupling, Kinetic equation, Hydrodynamic
             approximation, Diffusion approximation},
   Abstract = {This paper presents a general methodology to design
             macroscopic fluid models that take into account localized
             kinetic upscaling effects. The fluid models are solved in
             the whole domain together with a localized kinetic upscaling
             that corrects the fluid model wherever it is necessary. This
             upscaling is obtained by solving a kinetic equation on the
             nonequilibrium part of the distribution function. This
             equation is solved only locally and is related to the fluid
             equation through a downscaling effect. The method does not
             need to find an interface condition as do usual domain
             decomposition methods to match fluid and kinetic
             representations. We show our approach applies to problems
             that have a hydrodynamic time scale as well as to problems
             with diffusion time scale. Simple numerical schemes are
             proposed to discretize our models, and several numerical
             examples are used to validate the method. © 2006 Society
             for Industrial and Applied Mathematics.},
   Doi = {10.1137/060651574},
   Key = {fds246901}
}

@article{fds246957,
   Author = {Moore, J and Liu, J-G and Zhou, K and Yu, L},
   Title = {Effects of genotype and environment on the antioxidant
             properties of hard winter wheat bran},
   Journal = {Journal of Agricultural and Food Chemistry},
   Volume = {54},
   Number = {15},
   Pages = {5313-5322},
   Year = {2006},
   ISSN = {0021-8561},
   url = {http://dx.doi.org/10.1021/jf060381l},
   Abstract = {Recent consumer interest in controlling and preventing
             chronic diseases through improved diet has promoted research
             on the bioactive components of agricultural products. Wheat
             is an important agricultural and dietary commodity worldwide
             with known antioxidant properties concentrated mostly in the
             bran fraction. The objective of this study was to determine
             the relative contributions of genotype (G) and growing
             environment (E) to hard winter wheat bran antioxidant
             properties, as well as correlations of these properties to
             growing conditions. Bran samples of 20 hard winter wheat
             varieties grown in two locations were examined for their
             free radical scavenging capacities against DPPH, ABTS
             cation, peroxyl (ORAC), and superoxide anion radicals and
             chelating properties, as well as their total phenolics and
             phenolic acid compositions. Results showed significant
             differences for all antioxidant properties tested and
             multiple significant correlations between these properties.
             A factorial designed analysis of variance for these data and
             pooled previously published data showed similar results for
             four of the six antioxidant properties, indicating that G
             effects were considerably larger than E effects for
             chelating capacity and DPPH radical scavenging properties,
             whereas E was much stronger than G for ABTS cation radical
             scavenging capacity and total phenolics, although small
             interaction effects (G x E) were significant for all
             antioxidant properties analyzed. Results also showed
             significant correlations between temperature stress or solar
             radiation and some antioxidant properties. These results
             indicate that each antioxidant property of hard winter wheat
             bran is influenced differently by genotype and growing
             conditions. © 2006 American Chemical Society.},
   Doi = {10.1021/jf060381l},
   Key = {fds246957}
}

@article{fds246960,
   Author = {Liu, J-G and Wang, W-C},
   Title = {Convergence analysis of the energy and helicity preserving
             scheme for axisymmetric flows},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {44},
   Number = {6},
   Pages = {2456-2480},
   Year = {2006},
   ISSN = {0036-1429},
   url = {http://dx.doi.org/10.1137/050639314},
   Abstract = {We give an error estimate for the energy and helicity
             preserving scheme (EHPS) in second order finite difference
             setting on axisymmetric incompressible flows with swirling
             velocity. This is accomplished by a weighted energy
             estimate, along with careful and nonstandard local
             truncation error analysis near the geometric singularity and
             a far field decay estimate for the stream function. A key
             ingredient in our a priori estimate is the permutation
             identities associated with the Jacobians, which are also a
             unique feature that distinguishes EHPS from standard finite
             difference schemes. © 2006 Society for Industrial and
             Applied Mathematics.},
   Doi = {10.1137/050639314},
   Key = {fds246960}
}

@article{fds246964,
   Author = {Liu, J-G and Samelson, R and Wang, C},
   Title = {Global weak solution of planetary geostrophic equations with
             inviscid geostrophic balance},
   Journal = {Applicable Analysis},
   Volume = {85},
   Pages = {593-605},
   Year = {2006},
   Key = {fds246964}
}

@article{fds246877,
   Author = {Simsek, E and Liu, J and Liu, QH},
   Title = {A spectral integral method for layered media},
   Journal = {International Symposium on Antennas and Propagation
             (APSURSI)},
   Volume = {3 B},
   Pages = {238-},
   Year = {2005},
   ISSN = {1522-3965},
   url = {http://dx.doi.org/10.1109/APS.2005.1552481},
   Abstract = {In order to solve layered-medium problems in areas such as
             interconnect simulations and subsurface sensing, various
             numerical methods such as finite difference time domain
             methods, finite element methods, the method of moments
             (MoM), and the fast multipole method (FMM), have been
             developed. In this work, we are concerned with piecewise
             homogeneous objects embedded in a layered medium. As such,
             the surface integral equation (SIE) can be used to reduce
             the number of the unknowns compared with the volume integral
             equation. The SIE has been solved first with MOM and then
             with FMM to calculate the scattered electromagnetic fields
             from a homogeneous scatter with arbitrary geometry in a free
             space. Liu et.al. developed a spectral integral method (SIM)
             as an alternative way of solving the surface integral
             equation more efficiently than MOM (J. Liu and Q. H. Liu,
             IEEE Micro, and Wire. Comp. Lett., 14, 3, 97-99, 2004) for
             arbitrarily-shaped smooth dielectric cylinders in a free
             space. In this work we extend this method to
             arbitrarily-shaped smooth perfect electrical conductor (PEC)
             and dielectric cylinders in a multilayer medium. The main
             ingredients of this method are the use of fast Fourier
             transform (FFT) algorithm and the subtraction of
             singularities in Green's functions to achieve a spectral
             accuracy in the integral. 2D Green's functions for layered
             media are computed via numerical integration of a Sommerfeld
             type integral. To obtain spectral accuracy in the SIM, it is
             important that the Sommerfeld integration is obtained with
             high precision through singularity subtraction. We have
             demonstrated the spectral accuracy of this method and the
             reduced computational cost from the MOM. This method can
             also be extended to three dimensions. ©2005
             IEEE.},
   Doi = {10.1109/APS.2005.1552481},
   Key = {fds246877}
}

@article{fds246874,
   Author = {Liu, J and Liu, QH},
   Title = {A spectral integral method for periodic and nonperiodic
             structures},
   Journal = {International Symposium on Antennas and Propagation
             (APSURSI)},
   Volume = {4},
   Pages = {3875-3878},
   Year = {2004},
   ISSN = {1522-3965},
   Abstract = {A spectral integral method for periodic and non periodic
             structures by surface integral equation method was
             discussed. The boundary integral equations outside and
             inside the scatter for a dielectric object was presented.
             The method was based on fast fourier transform (FFT)
             algorithm and the subtraction of sigularities in Green's
             fuctions. The method demonstrated high accuracy and reduced
             computational cost and the method was extended to three
             dimensions.},
   Key = {fds246874}
}

@article{fds246876,
   Author = {Liu, J and Liu, QH},
   Title = {A Spectral Integral Method (SIM) for Periodic and
             Nonperiodic Structures},
   Journal = {IEEE Microwave and Wireless Components Letters},
   Volume = {14},
   Number = {3},
   Pages = {97-99},
   Year = {2004},
   url = {http://dx.doi.org/10.1109/LMWC.2004.824806},
   Abstract = {This letter presents a spectral integral method for
             electromagnetic scattering from dielectric objects with
             closed boundary. The method is developed for both periodic
             and non-periodic structures. Numerical results demonstrate
             the spectral accuracy of the method, and the advantages over
             the method of moments.},
   Doi = {10.1109/LMWC.2004.824806},
   Key = {fds246876}
}

@article{fds246902,
   Author = {Liu, J-G and Wang, W-C},
   Title = {Energy and helicity preserving schemes for hydro- and
             magnetohydro-dynamics flows with symmetry},
   Journal = {Journal of Computational Physics},
   Volume = {200},
   Number = {1},
   Pages = {8-33},
   Year = {2004},
   url = {http://dx.doi.org/10.1016/j.jcp.2004.03.005},
   Abstract = {We propose a class of simple and efficient numerical scheme
             for incompressible fluid equations with coordinate symmetry.
             By introducing a generalized vorticity-stream formulation,
             the divergence free constraints are automatically satisfied.
             In addition, with explicit treatment of the nonlinear terms
             and local vorticity boundary condition, the Navier-Stokes
             (MHD, respectively) equation essentially decouples into 2
             (4, respectively) scalar equation and thus the scheme is
             very efficient. Moreover, with proper discretization of the
             nonlinear terms, the scheme preserves both energy and
             helicity identities numerically. This is achieved by
             recasting the nonlinear terms (convection, vorticity
             stretching, geometric source, Lorentz force and
             electro-motive force) in terms of Jacobians. This
             conservative property is valid even in the presence of the
             pole singularity for axisymmetric flows. The exact
             conservation of energy and helicity has effectively
             eliminated excessive numerical viscosity. Numerical examples
             have demonstrated both accuracy and efficiency of the
             scheme. Finally, local mesh refinement near the boundary can
             also be easily incorporated into the scheme without extra
             cost. © 2004 Elsevier Inc. All rights reserved.},
   Doi = {10.1016/j.jcp.2004.03.005},
   Key = {fds246902}
}

@article{fds246954,
   Author = {Lin, H-E and Liu, J-G and Xu, W-Q},
   Title = {Effects of small viscosity and far field boundary conditions
             for hyperbolic systems},
   Journal = {Communications on Pure and Applied Analysis},
   Volume = {3},
   Number = {2},
   Pages = {267-290},
   Year = {2004},
   ISSN = {1534-0392},
   Abstract = {In this paper we study the effects of small viscosity term
             and the far-field boundary conditions for systems of
             convection-diffusion equations in the zero viscosity limit.
             The far-field boundary conditions are classified and the
             corresponding solution structures are analyzed. It is
             confirmed that the Neumann type of far-field boundary
             condition is preferred. On the other hand, we also identify
             a class of improperly coupled boundary conditions which lead
             to catastrophic reflection waves dominating the inlet in the
             zero viscosity limit. The analysis is performed on the
             linearized convection-diffusion model which well describes
             the behavior at the far field for many physical and
             engineering systems such as fluid dynamical equations and
             electro-magnetic equations. The results obtained here should
             provide some theoretical guidance for designing effective
             far field boundary conditions.},
   Key = {fds246954}
}

@article{fds246955,
   Author = {Liu, J-G and Xu, W-Q},
   Title = {Far field boundary condition for convection diffusion
             equation at zero viscosity limit},
   Journal = {Quarterly of Applied Mathematics},
   Volume = {62},
   Number = {1},
   Pages = {27-52},
   Year = {2004},
   Abstract = {In this paper, we give a systematic study of the boundary
             layer behavior for linear convection-diffusion equation in
             the zero viscosity limit. We analyze the boundary layer
             structures in the viscous solution and derive the boundary
             condition satisfied by the viscosity limit as a solution of
             the inviscid equation. The results confirm that the Neumann
             type of far-field boundary condition is preferred in the
             outlet and characteristic boundary dondition. Under some
             appropriate regularity and compatibility conditions on the
             initial and boundary data, we obtain optimal error estimates
             between the full viscous solution and the inviscid solution
             with suitable boundary layer corrections. These results hold
             in arbitrary space dimensions and similar statements also
             hold for the strip problem This model well describes the
             behavior at the far-field for many physical and engineering
             systems such as fluid dynamical equation and
             electro-magnetic equation. The results obtained here should
             provide some theoretical guidance for designing effective
             far-field boundary conditions.},
   Key = {fds246955}
}

@article{fds246956,
   Author = {Wang, C and Liu, J-G and Johnston, H},
   Title = {Analysis of a fourth order finite difference method for the
             incompressible Boussinesq equations},
   Journal = {Numerische Mathematik},
   Volume = {97},
   Number = {3},
   Pages = {555-594},
   Year = {2004},
   url = {http://dx.doi.org/10.1007/s00211-003-0508-3},
   Abstract = {The convergence of a fourth order finite difference method
             for the 2-D unsteady, viscous incompressible Boussinesq
             equations, based on the vorticity-stream function
             formulation, is established in this article. A compact
             fourth order scheme is used to discretize the momentum
             equation, and long-stencil fourth order operators are
             applied to discretize the temperature transport equation. A
             local vorticity boundary condition is used to enforce the
             no-slip boundary condition for the velocity. One-sided
             extrapolation is used near the boundary, dependent on the
             type of boundary condition for the temperature, to prescribe
             the temperature at "ghost" points lying outside of the
             computational domain. Theoretical results of the stability
             and accuracy of the method are also provided. In numerical
             experiments the method has been shown to be capable of
             producing highly resolved solutions at a reasonable
             computational cost.},
   Doi = {10.1007/s00211-003-0508-3},
   Key = {fds246956}
}

@article{fds246959,
   Author = {Li, B and Liu, J-G},
   Title = {Eptaxial growth without slope selection: energetics,
             coarsening, and dynamic scaling},
   Journal = {J. Nonlinear Sci.},
   Volume = {14},
   Number = {5},
   Pages = {429-451},
   Year = {2004},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-004-0634-9},
   Abstract = {We study a continuum model for epitaxial growth of thin
             films in which the slope of mound structure of film surface
             increases. This model is a diffusion equation for the
             surface height profile h which is assumed to satisfy the
             periodic boundary condition. The equation happens to possess
             a Liapunov or "free-energy" functional. This functional
             consists of the term |Δ h| 2, which represents the
             surface diffusion, and-log (1 + |∇ h| 2), which
             describes the effect of kinetic asymmetry in the adatom
             attachment-detachment. We first prove for large time t that
             the interface width-the standard deviation of the height
             profile-is bounded above by O(t 1/2), the averaged gradient
             is bounded above by O(t 1/4), and the averaged energy is
             bounded below by O(-log t). We then consider a small
             coefficient ε 2 of |Δ h| 2 with ε = 1/L and L the
             linear size of the underlying system, and study the energy
             asymptotics in the large system limit ε → 0. We
             show that global minimizers of the free-energy functional
             exist for each ε > 0, the L 2-norm of the gradient of
             any global minimizer scales as O(1/ε), and the global
             minimum energy scales as O( log ε). The existence of
             global energy minimizers and a scaling argument are used to
             construct a sequence of equilibrium solutions with different
             wavelengths. Finally, we apply our minimum energy estimates
             to derive bounds in terms of the linear system size L for
             the saturation interface width and the corresponding
             saturation time. © 2005 Springer.},
   Doi = {10.1007/s00332-004-0634-9},
   Key = {fds246959}
}

@article{fds246962,
   Author = {Johnston, H and Liu, J-G},
   Title = {Accurate, stable and efficient Navier-Stokes solvers based
             on explicit treatment of the pressure term},
   Journal = {Journal of Computational Physics},
   Volume = {199},
   Number = {1},
   Pages = {221-259},
   Year = {2004},
   url = {http://dx.doi.org/10.1016/j.jcp.2004.02.009},
   Abstract = {We present numerical schemes for the incompressible
             Navier-Stokes equations based on a primitive variable
             formulation in which the incompressibility constraint has
             been replaced by a pressure Poisson equation. The pressure
             is treated explicitly in time, completely decoupling the
             computation of the momentum and kinematic equations. The
             result is a class of extremely efficient Navier-Stokes
             solvers. Full time accuracy is achieved for all flow
             variables. The key to the schemes is a Neumann boundary
             condition for the pressure Poisson equation which enforces
             the incompressibility condition for the velocity field.
             Irrespective of explicit or implicit time discretization of
             the viscous term in the momentum equation the explicit time
             discretization of the pressure term does not affect the time
             step constraint. Indeed, we prove unconditional stability of
             the new formulation for the Stokes equation with explicit
             treatment of the pressure term and first or second order
             implicit treatment of the viscous term. Systematic numerical
             experiments for the full Navier-Stokes equations indicate
             that a second order implicit time discretization of the
             viscous term, with the pressure and convective terms treated
             explicitly, is stable under the standard CFL condition.
             Additionally, various numerical examples are presented,
             including both implicit and explicit time discretizations,
             using spectral and finite difference spatial
             discretizations, demonstrating the accuracy, flexibility and
             efficiency of this class of schemes. In particular, a
             Galerkin formulation is presented requiring only C0 elements
             to implement. © 2004 Elsevier Inc. All rights
             reserved.},
   Doi = {10.1016/j.jcp.2004.02.009},
   Key = {fds246962}
}

@article{fds246963,
   Author = {Ghil, M and Liu, J-G and Wang, C and Wang, S},
   Title = {Boundary-layer separation and adverse pressure gradient for
             2-D viscous incompressible flow},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {197},
   Number = {1-2},
   Pages = {149-173},
   Year = {2004},
   ISSN = {0167-2789},
   url = {http://dx.doi.org/10.1016/j.physd.2004.06.012},
   Abstract = {We study the detailed process of bifurcation in the flow's
             topological structure for a two-dimensional (2-D)
             incompressible flow subject to no-slip boundary conditions
             and its connection with boundary-layer separation. The
             boundary-layer separation theory of M. Ghil, T. Ma and S.
             Wang, based on the structural-bifurcation concept, is
             translated into vorticity form. The vorticily formulation of
             the theory shows that structural bifurcation occurs whenever
             a degenerate singular point for the vorticity appears on the
             boundary; this singular point is characterized by nonzero
             tangential second-order derivative and nonzero time
             derivative of the vorticity. Furthermore, we prove the
             presence of an adverse pressure gradient at the critical
             point, due to reversal in the direction of the pressure
             force with respect to the basic shear flow at this point. A
             numerical example of 2-D driven-cavity flow, governed by the
             Navier Stokes equations, is presented; boundary-layer
             separation occurs, the bifurcation criterion is satisfied,
             and an adverse pressure gradient is shown to be present. ©
             2004 Elsevier B.V. All rights reserved.},
   Doi = {10.1016/j.physd.2004.06.012},
   Key = {fds246963}
}

@article{fds246965,
   Author = {Liu, J-G and Wang, C},
   Title = {High order finite difference method for unsteady
             incompressible flow on multi-connected domain in
             vorticity-stream function formulation},
   Journal = {Computer and Fluids},
   Volume = {33},
   Number = {2},
   Pages = {223-255},
   Year = {2004},
   url = {http://dx.doi.org/10.1016/S0045-7930(03)00037-9},
   Abstract = {Using the vorticity and stream function variables is an
             effective way to compute 2-D incompressible flow due to the
             facts that the incompressibility constraint for the velocity
             is automatically satisfied, the pressure variable is
             eliminated, and high order schemes can be efficiently
             implemented. However, a difficulty arises in a
             multi-connected computational domain in determining the
             constants for the stream function on the boundary of the
             "holes". This is an especially challenging task for the
             calculation of unsteady flows, since these constants vary
             with time to reflect the total fluxes of the flow in each
             sub-channel. In this paper, we propose an efficient method
             in a finite difference setting to solve this problem and
             present some numerical experiments, including an accuracy
             check of a Taylor vortex-type flow, flow past a
             non-symmetric square, and flow in a heat exchanger. ©
             2003 Elsevier Ltd. All rights reserved.},
   Doi = {10.1016/S0045-7930(03)00037-9},
   Key = {fds246965}
}

@article{fds304583,
   Author = {Liu, J-G and Wang, C},
   Title = {High order finite difference methods for unsteady
             incompressible flows in multi-connected domains},
   Journal = {Computers and Fluids},
   Volume = {33},
   Number = {2},
   Pages = {223-255},
   Year = {2004},
   url = {http://dx.doi.org/10.1016/S0045-7930(03)00037-9},
   Abstract = {Using the vorticity and stream function variables is an
             effective way to compute 2-D incompressible flow due to the
             facts that the incompressibility constraint for the velocity
             is automatically satisfied, the pressure variable is
             eliminated, and high order schemes can be efficiently
             implemented. However, a difficulty arises in a
             multi-connected computational domain in determining the
             constants for the stream function on the boundary of the
             "holes". This is an especially challenging task for the
             calculation of unsteady flows, since these constants vary
             with time to reflect the total fluxes of the flow in each
             sub-channel. In this paper, we propose an efficient method
             in a finite difference setting to solve this problem and
             present some numerical experiments, including an accuracy
             check of a Taylor vortex-type flow, flow past a
             non-symmetric square, and flow in a heat exchanger. © 2003
             Elsevier Ltd. All rights reserved.},
   Doi = {10.1016/S0045-7930(03)00037-9},
   Key = {fds304583}
}

@article{fds304585,
   Author = {Li, B and Liu, J-G},
   Title = {Epitaxial growth without slope selection: Energetics,
             coarsening, and dynamic scaling},
   Journal = {Journal of Nonlinear Science},
   Volume = {14},
   Number = {5},
   Pages = {429-451},
   Year = {2004},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-004-0634-9},
   Abstract = {We study a continuum model for epitaxial growth of thin
             films in which the slope of mound structure of film surface
             increases. This model is a diffusion equation for the
             surface height profile h which is assumed to satisfy the
             periodic boundary condition. The equation happens to possess
             a Liapunov or "free-energy" functional. This functional
             consists of the term |Δ h| 2, which represents the surface
             diffusion, and-log (1 + |∇ h| 2), which describes the
             effect of kinetic asymmetry in the adatom
             attachment-detachment. We first prove for large time t that
             the interface width-the standard deviation of the height
             profile-is bounded above by O(t 1/2), the averaged gradient
             is bounded above by O(t 1/4), and the averaged energy is
             bounded below by O(-log t). We then consider a small
             coefficient ε 2 of |Δ h| 2 with ε = 1/L and L the linear
             size of the underlying system, and study the energy
             asymptotics in the large system limit ε → 0. We show that
             global minimizers of the free-energy functional exist for
             each ε > 0, the L 2-norm of the gradient of any global
             minimizer scales as O(1/ε), and the global minimum energy
             scales as O( log ε). The existence of global energy
             minimizers and a scaling argument are used to construct a
             sequence of equilibrium solutions with different
             wavelengths. Finally, we apply our minimum energy estimates
             to derive bounds in terms of the linear system size L for
             the saturation interface width and the corresponding
             saturation time. © 2005 Springer.},
   Doi = {10.1007/s00332-004-0634-9},
   Key = {fds304585}
}

@article{fds246950,
   Author = {Wang, C and Liu, J-G},
   Title = {Fourth order convergence of a compact difference solver for
             incompressible flow},
   Journal = {Commun. Appl. Anal.},
   Volume = {7},
   Pages = {171-191},
   Year = {2003},
   Key = {fds246950}
}

@article{fds246951,
   Author = {Wang, C and Liu, J-G},
   Title = {Positivity property of second-order flux-splitting schemes
             for the compressible Euler equations},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {3},
   Number = {2},
   Pages = {201-228},
   Year = {2003},
   Abstract = {A class of upwind flux splitting methods in the Euler
             equations of compressible flow is considered in this paper.
             Using the property that Euler flux F(U) is a homogeneous
             function of degree one in U, we reformulate the splitting
             fluxes with F+ = A+U, F- = A -U, and the corresponding
             matrices are either symmetric or symmetrizable and keep only
             non-negative and non-positive eigenvalues. That leads to the
             conclusion that the first order schemes are positive in the
             sense of Lax-Liu [18], which implies that it is L2- stable
             in some suitable sense. Moreover, the second order scheme is
             a stable perturbation of the first order scheme, so that the
             positivity of the second order schemes is also established,
             under a CFL-like condition. In addition, these splitting
             methods preserve the positivity of density and
             energy.},
   Key = {fds246951}
}

@article{fds246952,
   Author = {Chainais-Hillairet, C and Liu, J-G and Peng, Y-J},
   Title = {Finite volume scheme for multi-dimensional drift-diffusion
             equations and convergence analysis},
   Journal = {Mathematical Modelling and Numerical Analysis},
   Volume = {37},
   Number = {2},
   Pages = {319-338},
   Year = {2003},
   Abstract = {We introduce a finite volume scheme for multi-dimensional
             drift-diffusion equations. Such equations arise from the
             theory of semiconductors and are composed of two continuity
             equations coupled with a Poisson equation. In the case that
             the continuity equations are non degenerate, we prove the
             convergence of the scheme and then the existence of
             solutions to the problem. The key point of the proof relies
             on the construction of an approximate gradient of the
             electric potential which allows us to deal with coupled
             terms in the continuity equations. Finally, a numerical
             example is given to show the efficiency of the
             scheme.},
   Key = {fds246952}
}

@article{fds246953,
   Author = {Duraisamy, K and Baeder, JD and Liu, J-G},
   Title = {Concepts and Application of Time-Limiters to High Resolution
             Schemes},
   Journal = {Journal of Scientific Computing},
   Volume = {19},
   Number = {1-3},
   Pages = {139-162},
   Year = {2003},
   ISSN = {0885-7474},
   url = {http://dx.doi.org/10.1023/A:1025395707090},
   Abstract = {A new class of implicit high-order non-oscillatory time
             integration schemes is introduced in a method-of-lines
             framework. These schemes can be used in conjunction with an
             appropriate spatial discretization scheme for the numerical
             solution of time dependent conservation equations. The main
             concept behind these schemes is that the order of accuracy
             in time is dropped locally in regions where the time
             evolution of the solution is not smooth. By doing this, an
             attempt is made at locally satisfying monotonicity
             conditions, while maintaining a high order of accuracy in
             most of the solution domain. When a linear high order time
             integration scheme is used along with a high order spatial
             discretization, enforcement of monotonicity imposes severe
             time-step restrictions. We propose to apply limiters to
             these time-integration schemes, thus making them non-linear.
             When these new schemes are used with high order spatial
             discretizations, solutions remain non-oscillatory for much
             larger time-steps as compared to linear time integration
             schemes. Numerical results obtained on scalar conservation
             equations and systems of conservation equations are highly
             promising.},
   Doi = {10.1023/A:1025395707090},
   Key = {fds246953}
}

@article{fds246961,
   Author = {Weinan, E and Liu, J-G},
   Title = {Gauge method for viscous incompressible flows},
   Journal = {Comm. Math. Sci.},
   Volume = {1},
   Pages = {317-332},
   Year = {2003},
   Key = {fds246961}
}

@article{fds246966,
   Author = {Li, B and Liu, J-G},
   Title = {Thin film epitaxy with or without slope selection},
   Journal = {European Journal of Applied Mathematics},
   Volume = {14},
   Number = {6},
   Pages = {713-743},
   Year = {2003},
   url = {http://dx.doi.org/10.1017/S095679250300528X},
   Abstract = {Two nonlinear diffusion equations for thin film epitaxy,
             with or without slope selection, are studied in this work.
             The nonlinearity models the Ehrlich-Schwoebel effect - the
             kinetic asymmetry in attachment and detachment of adatoms to
             and from terrace boundaries. Both perturbation analysis and
             numerical simulation are presented to show that such an
             atomistic effect is the origin of a nonlinear morphological
             instability, in a rough-smooth-rough pattern, that has been
             experimentally observed as transient in an early stage of
             epitaxial growth on rough surfaces. Initial-boundary-value
             problems for both equations are proven to be well-posed, and
             the solution regularity is also obtained. Galerkin spectral
             approximations are studied to provide both a priori bounds
             for proving the well-posedness and numerical schemes for
             simulation. Numerical results are presented to confirm part
             of the analysis and to explore the difference between the
             two models on coarsening dynamics.},
   Doi = {10.1017/S095679250300528X},
   Key = {fds246966}
}

@article{fds246967,
   Author = {Chern, I-L and Liu, J-G and Wang, W-C},
   Title = {Accurate evaluation of electrostatics for macromolecules in
             solution},
   Journal = {Methods and Applications of Analysis},
   Volume = {10},
   Pages = {309-328},
   Year = {2003},
   Key = {fds246967}
}

@article{fds246968,
   Author = {Liu, J-G and Wang, C and Johnston, H},
   Title = {A Fourth Order Scheme for Incompressible Boussinesq
             Equations},
   Journal = {Journal of Scientific Computing},
   Volume = {18},
   Number = {2},
   Pages = {253-285},
   Year = {2003},
   ISSN = {0885-7474},
   url = {http://dx.doi.org/10.1023/A:1021168924020},
   Abstract = {A fourth order finite difference method is presented for the
             2D unsteady viscous incompressible Boussinesq equations in
             vorticity-stream function formulation. The method is
             especially suitable for moderate to large Reynolds number
             flows. The momentum equation is discretized by a compact
             fourth order scheme with the no-slip boundary condition
             enforced using a local vorticity boundary condition. Fourth
             order long-stencil discretizations are used for the
             temperature transport equation with one-sided extrapolation
             applied near the boundary. The time stepping scheme for both
             equations is classical fourth order Runge-Kutta. The method
             is highly efficient. The main computation consists of the
             solution of two Poisson-like equations at each Runge-Kutta
             time stage for which standard FFT based fast Poisson solvers
             are used. An example of Lorenz flow is presented, in which
             the full fourth order accuracy is checked. The numerical
             simulation of a strong shear flow induced by a temperature
             jump, is resolved by two perfectly matching resolutions.
             Additionally, we present benchmark quality simulations of a
             differentially-heated cavity problem. This flow was the
             focus of a special session at the first MIT conference on
             Computational Fluid and Solid Mechanics in June
             2001.},
   Doi = {10.1023/A:1021168924020},
   Key = {fds246968}
}

@article{fds246937,
   Author = {Wang, C and Liu, J-G},
   Title = {Analysis of finite difference schemes for unsteady
             Navier-Stokes equations in vorticity formulation},
   Journal = {Numerische Mathematik},
   Volume = {91},
   Number = {3},
   Pages = {543-576},
   Year = {2002},
   url = {http://dx.doi.org/10.1007/s002110100311},
   Abstract = {In this paper, we provide stability and convergence analysis
             for a class of finite difference schemes for unsteady
             incompressible Navier-Stokes equations in vorticity-stream
             function formulation. The no-slip boundary condition for the
             velocity is converted into local vorticity boundary
             conditions. Thorn's formula, Wilkes' formula, or other local
             formulas in the earlier literature can be used in the second
             order method; while high order formulas, such as Briley's
             formula, can be used in the fourth order compact difference
             scheme proposed by E and Liu. The stability analysis of
             these long-stencil formulas cannot be directly derived from
             straightforward manipulations since more than one interior
             point is involved in the formula. The main idea of the
             stability analysis is to control local terms by global
             quantities via discrete elliptic regularity for stream
             function. We choose to analyze the second order scheme with
             Wilkes' formula in detail. In this case, we can avoid the
             complicated technique necessitated by the Strang-type high
             order expansions. As a consequence, our analysis results in
             almost optimal regularity assumption for the exact solution.
             The above methodology is very general. We also give a
             detailed analysis for the fourth order scheme using a 1-D
             Stokes model.},
   Doi = {10.1007/s002110100311},
   Key = {fds246937}
}

@article{fds246938,
   Author = {Weinan, E and Liu, J-G},
   Title = {Projection method III: Spatial discretization on the
             staggered grid},
   Journal = {Mathematics of Computation},
   Volume = {71},
   Number = {237},
   Pages = {27-47},
   Year = {2002},
   url = {http://dx.doi.org/10.1090/S0025-5718-01-01313-8},
   Abstract = {In E & Liu (SIAM J Numer. Anal., 1995), we studied
             convergence and the structure of the error for several
             projection methods when the spatial variable was kept
             continuous (we call this the semi-discrete case). In this
             paper, we address similar questions for the fully discrete
             case when the spatial variables are discretized using a
             staggered grid. We prove that the numerical solution in
             velocity has full accuracy up to the boundary, despite the
             fact that there are numerical boundary layers present in the
             semi-discrete solutions.},
   Doi = {10.1090/S0025-5718-01-01313-8},
   Key = {fds246938}
}

@article{fds246939,
   Author = {Johnston, H and Liu, J-G},
   Title = {Finite difference schemes for incompressible flow based on
             local pressure boundary conditions},
   Journal = {Journal of Computational Physics},
   Volume = {180},
   Number = {1},
   Pages = {120-154},
   Year = {2002},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1006/jcph.2002.7079},
   Abstract = {In this paper we discuss the derivation and use of local
             pressure boundary conditions for finite difference schemes
             for the unsteady incompressible Navier-Stokes equations in
             the velocity-pressure formulation. Their use is especially
             well suited for the computation of moderate to large
             Reynolds number flows. We explore the similarities between
             the implementation and use of local pressure boundary
             conditions and local vorticity boundary conditions in the
             design of numerical schemes for incompressible flow in 2D.
             In their respective formulations, when these local numerical
             boundary conditions are coupled with a fully explicit
             convectively stable time stepping procedure, the resulting
             methods are, simple to implement and highly efficient.
             Unlike the vorticity formulation, the use of the local
             pressure boundary condition approach is readily applicable
             to 3D flows. The simplicity of the local pressure boundary
             condition approach and its easy adaptation to more general
             flow settings make the resulting scheme an attractive
             alternative to the more popular methods for solving the
             Navier-Stokes equations in the velocity-pressure
             formulation. We present numerical results of a second-order
             finite difference scheme on a nonstaggered grid using local
             pressure boundary conditions. Stability and accuracy of the
             scheme applied to Stokes flow is demonstrated using normal
             mode analysis. Also described is the extension of the method
             to variable density flows. © 2002 Elsevier Science
             (USA).},
   Doi = {10.1006/jcph.2002.7079},
   Key = {fds246939}
}

@article{fds304582,
   Author = {Liu, J-G and Xin, Z},
   Title = {Convergence of the point vortex method for 2-D vortex
             sheet},
   Journal = {Mathematics of Computation},
   Volume = {70},
   Number = {234},
   Pages = {595-606},
   Year = {2001},
   url = {http://dx.doi.org/10.1090/S0025-5718-00-01271-0},
   Abstract = {We give an elementary proof of the convergence of the point
             vortex method (PVM) to a classical weak solution for the
             two-dimensional incompressible Euler equations with initial
             vorticity being a finite Radon measure of distinguished sign
             and the initial velocity of locally bounded energy. This
             includes the important example of vortex sheets, which
             exhibits the classical Kelvin-Helmholtz instability. A
             surprise fact is that although the velocity fields generated
             by the point vortex method do not have bounded local kinetic
             energy, the limiting velocity field is shown to have a
             bounded local kinetic energy.},
   Doi = {10.1090/S0025-5718-00-01271-0},
   Key = {fds304582}
}

@article{fds246873,
   Author = {Liu, J-G and Weinan, E},
   Title = {Simple finite element method in vorticity formulation for
             incompressible flows},
   Journal = {Mathematics of Computation},
   Volume = {70},
   Number = {234},
   Pages = {579-593},
   Year = {2001},
   url = {http://dx.doi.org/10.1090/S0025-5718-00-01239-4},
   Abstract = {A very simple and efficient finite element method is
             introduced for two and three dimensional viscous
             incompressible flows using the vorticity formulation. This
             method relies on recasting the traditional finite element
             method in the spirit of the high order accurate finite
             difference methods introduced by the authors in another
             work. Optimal accuracy of arbitrary order can be achieved
             using standard finite element or spectral elements. The
             method is convectively stable and is particularly suited for
             moderate to high Reynolds number flows.},
   Doi = {10.1090/S0025-5718-00-01239-4},
   Key = {fds246873}
}

@article{fds246934,
   Author = {Liu, J-G and Wang, W-C},
   Title = {An energy-preserving MAC-Yee scheme for the incompressible
             MHD equation},
   Journal = {Journal of Computational Physics},
   Volume = {174},
   Number = {1},
   Pages = {12-37},
   Year = {2001},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1006/jcph.2001.6772},
   Abstract = {We propose a simple and efficient finite-difference method
             for the incompressible MHD equation. The numerical method
             combines the advantage of the MAC scheme for the
             Navier-Stokes equation and Yee's scheme for the Maxwell
             equation. In particular, the semi-discrete version of our
             scheme introduces no numerical dissipation and preserves the
             energy identity exactly. © 2001 Elsevier
             Science.},
   Doi = {10.1006/jcph.2001.6772},
   Key = {fds246934}
}

@article{fds246935,
   Author = {Liu, J-G and Weinan, E},
   Title = {Simple finite element method in vorticity formulation for
             incompressible flow},
   Journal = {Math. Comp.},
   Volume = {69},
   Pages = {1385-1407},
   Year = {2001},
   Key = {fds246935}
}

@article{fds246936,
   Author = {Liu, J-G and Xin, Z},
   Title = {Convergence of point vortex method for 2-D vortex
             sheet},
   Journal = {Math. Comp.},
   Volume = {70},
   Number = {234},
   Pages = {565-606},
   Year = {2001},
   url = {http://dx.doi.org/10.1090/S0025-5718-00-01271-0},
   Abstract = {We give an elementary proof of the convergence of the point
             vortex method (PVM) to a classical weak solution for the
             two-dimensional incompressible Euler equations with initial
             vorticity being a finite Radon measure of distinguished sign
             and the initial velocity of locally bounded energy. This
             includes the important example of vortex sheets, which
             exhibits the classical Kelvin-Helmholtz instability. A
             surprise fact is that although the velocity fields generated
             by the point vortex method do not have bounded local kinetic
             energy, the limiting velocity field is shown to have a
             bounded local kinetic energy.},
   Doi = {10.1090/S0025-5718-00-01271-0},
   Key = {fds246936}
}

@article{fds246930,
   Author = {Liu, J-G and Xin, Z},
   Title = {Convergence of a Galerkin method for 2-D discontinuous Euler
             flows},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {53},
   Number = {6},
   Pages = {786-798},
   Year = {2000},
   Abstract = {We prove the convergence of a discontinuous Galerkin method
             approximating the 2-D incompressible Euler equations with
             discontinuous initial vorticity: ω0 ∈ L2(Ω).
             Furthermore, when ω0 ∈ L∞(Ω), the whole sequence is
             shown to be strongly convergent. This is the first
             convergence result in numerical approximations of this
             general class of discontinuous flows. Some important flows
             such as vortex patches belong to this class. © 2000 John
             Wiley & Sons, Inc.},
   Key = {fds246930}
}

@article{fds246931,
   Author = {Liu, J-G and Shu, C-W},
   Title = {A High-Order Discontinuous Galerkin Method for 2D
             Incompressible Flows},
   Journal = {Journal of Computational Physics},
   Volume = {160},
   Number = {2},
   Pages = {577-596},
   Year = {2000},
   url = {http://dx.doi.org/10.1006/jcph.2000.6475},
   Abstract = {In this paper we introduce a high-order discontinuous
             Galerkin method for two-dimensional incompressible flow in
             the vorticity stream-function formulation. The momentum
             equation is treated explicitly, utilizing the efficiency of
             the discontinuous Galerkin method. The stream function is
             obtained by a standard Poisson solver using continuous
             finite elements. There is a natural matching between these
             two finite element spaces, since the normal component of the
             velocity field is continuous across element boundaries. This
             allows for a correct upwinding gluing in the discontinuous
             Galerkin framework, while still maintaining total energy
             conservation with no numerical dissipation and total
             enstrophy stability. The method is efficient for inviscid or
             high Reynolds number flows. Optimal error estimates are
             proved and verified by numerical experiments. © 2000
             Academic Press.},
   Doi = {10.1006/jcph.2000.6475},
   Key = {fds246931}
}

@article{fds246932,
   Author = {Wang, C and Liu, J-G},
   Title = {Convergence of gauge method for incompressible
             flow},
   Journal = {Mathematics of Computation},
   Volume = {69},
   Number = {232},
   Pages = {1385-1407},
   Year = {2000},
   Abstract = {A new formulation, a gauge formulation of the incompressible
             Navier-Stokes equations in terms of an auxiliary field a and
             a gauge variable φ, u = a + ∇φ, was proposed recently by
             E and Liu. This paper provides a theoretical analysis of
             their formulation and verifies the computational advantages.
             We discuss the implicit gauge method, which uses backward
             Euler or Crank-Nicolson in time discretization. However, the
             boundary conditions for the auxiliary field a are
             implemented explicitly (vertical extrapolation). The
             resulting momentum equation is decoupled from the kinematic
             equation, and the computational cost is reduced to solving a
             standard heat and Poisson equation. Moreover, such explicit
             boundary conditions for the auxiliary field a will be shown
             to be unconditionally stable for Stokes equations. For the
             full nonlinear Navier-Stokes equations the time stepping
             constraint is reduced to the standard CFL constraint Δt/Δx
             ≤ C. We also prove first order convergence of the gauge
             method when we use MAC grids as our spatial discretization.
             The optimal error estimate for the velocity field is also
             obtained.},
   Key = {fds246932}
}

@article{fds246933,
   Author = {Weinan, E and Liu, J-G},
   Title = {Gauge finite element method for incompressible
             flows},
   Journal = {International Journal for Numerical Methods in
             Fluids},
   Volume = {34},
   Number = {8},
   Pages = {701-710},
   Year = {2000},
   ISSN = {0271-2091},
   url = {http://dx.doi.org/10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B},
   Abstract = {A finite element method for computing viscous incompressible
             flows based on the gauge formulation introduced in [Weinan
             E. Liu J-G. Gauge method for viscous incompressible flows.
             Journal of Computational Physics (submitted)] is presented.
             This formulation replaces the pressure by a gauge variable.
             This new gauge variable is a numerical tool and differs from
             the standard gauge variable that arises from decomposing a
             compressible velocity field. It has the advantage that an
             additional boundary condition can be assigned to the gauge
             variable, thus eliminating the issue of a pressure boundary
             condition associated with the original primitive variable
             formulation. The computational task is then reduced to
             solving standard heat and Poisson equations, which are
             approximated by straightforward, piecewise linear (or
             higher-order) finite elements. This method can achieve
             high-order accuracy at a cost comparable with that of
             solving standard heat and Poisson equations. It is naturally
             adapted to complex geometry and it is much simpler than
             traditional finite elements methods for incompressible
             flows. Several numerical examples on both structured and
             unstructured grids are presented. Copyright © 2000 John
             Wiley &amp; Sons, Ltd.},
   Doi = {10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B},
   Key = {fds246933}
}

@article{fds246927,
   Author = {Lefloch, PG and Liu, J-G},
   Title = {Generalized monotone schemes, discrete paths of extrema, and
             discrete entropy conditions},
   Journal = {Mathematics of Computation},
   Volume = {68},
   Number = {227},
   Pages = {1025-1055},
   Year = {1999},
   Abstract = {Solutions of conservation laws satisfy the monotonicity
             property: the number of local extrema is a non-increasing
             function of time, and local maximum/minimum values
             decrease/increase monotonically in time. This paper
             investigates this property from a numerical standpoint. We
             introduce a class of fully discrete in space and time, high
             order accurate, difference schemes, called generalized
             monotone schemes. Convergence toward the entropy solution is
             proven via a new technique of proof, assuming that the
             initial data has a finite number of extremum values only,
             and the flux-function is strictly convex. We define discrete
             paths of extrema by tracking local extremum values in the
             approximate solution. In the course of the analysis we
             establish the pointwise convergence of the trace of the
             solution along a path of extremum. As a corollary, we obtain
             a proof of convergence for a MUSCL-type scheme that is
             second order accurate away from sonic points and
             extrema.},
   Key = {fds246927}
}

@article{fds246929,
   Author = {Wang, ZJ and Liu, JG and Childress, S},
   Title = {Connection between corner vortices and shear layer
             instability in flow past an ellipse},
   Journal = {Physics of Fluids},
   Volume = {11},
   Number = {9},
   Pages = {2446-2448},
   Year = {1999},
   Abstract = {We investigate, by numerical simulation, the shear layer
             instability associated with the outer layer of a spiral
             vortex formed behind an impulsively started thin ellipse.
             The unstable free shear layer undergoes a secondary
             instability. We connect this instability with the dynamics
             of corner vortices adjacent to the tip of the ellipse by
             observing that the typical turnover time of the corner
             vortex matches the period of the unstable mode in the shear
             layer. We suggest that the corner vortex acts as a signal
             generator, and produces periodic perturbation which triggers
             the instability. © 1999 American Institute of
             Physics.},
   Key = {fds246929}
}

@article{fds246925,
   Author = {Xu, E and Liu, J-G},
   Title = {Pricing of mortgage-backed securities with option-adjusted
             spread},
   Journal = {Managerial Finance},
   Volume = {24},
   Pages = {94-109},
   Year = {1998},
   Key = {fds246925}
}

@article{fds246926,
   Author = {Choi, H and Liu, J-G},
   Title = {The Reconstruction of Upwind Fluxes for Conservation Laws:
             Its Behavior in Dynamic and Steady State
             Calculations},
   Journal = {Journal of Computational Physics},
   Volume = {144},
   Number = {2},
   Pages = {237-256},
   Year = {1998},
   url = {http://dx.doi.org/10.1006/jcph.1998.5970},
   Abstract = {The Euler equation of compressible flows is solved by the
             finite volume method, where high order accuracy is achieved
             by the reconstruction of each component of upwind fluxes of
             a flux splitting using the biased averaging procedure.
             Compared to the solution reconstruction in Godunov-type
             methods, its implementation is simple and easy, and the
             computational complexity is relatively low. This approach is
             parameter-free and requires neither a Riemann solver nor
             field-by-field decomposition. The numerical results from
             both dynamic and steady state calculations demonstrate the
             accuracy and robustness of this approach. Some techniques
             for the acceleration of the convergence to the steady state
             are discussed, including multigrid and multistage
             Runge-Kutta time methods. © 1998 Academic
             Press.},
   Doi = {10.1006/jcph.1998.5970},
   Key = {fds246926}
}

@article{fds246922,
   Author = {Weinan, E and Liu, J-G},
   Title = {Finite Difference Methods for 3D Viscous Incompressible
             Flows in the Vorticity-Vector Potential Formulation on
             Nonstaggered Grids},
   Journal = {Journal of Computational Physics},
   Volume = {138},
   Number = {1},
   Pages = {57-82},
   Year = {1997},
   url = {http://dx.doi.org/10.1006/jcph.1997.5815},
   Abstract = {Simple, efficient, and accurate finite difference methods
             are introduced for 3D unsteady viscous incompressible flows
             in the vorticity-vector potential formulation on
             nonstaggered grids. Two different types of methods are
             discussed. They differ in the implementation of the normal
             component of the vorticity boundary condition and
             consequently the enforcement of the divergence free
             condition for vorticity. Both second-order and fourth-order
             accurate schemes are developed. A detailed accuracy test is
             performed, revealing the structure of the error and the
             effect of how the convective terms are discretized near the
             boundary. The influence of the divergence free condition for
             vorticity to the overall accuracy is studied. Results on the
             cubic driven cavity flow at Reynolds number 500 and 3200 are
             shown and compared with that of the MAC scheme. © 1997
             Academic Press.},
   Doi = {10.1006/jcph.1997.5815},
   Key = {fds246922}
}

@article{fds246923,
   Author = {Chen, G-Q and Liu, J-G},
   Title = {Convergence of difference schemes with high resolution for
             conservation laws},
   Journal = {Mathematics of Computation},
   Volume = {66},
   Number = {219},
   Pages = {1027-1053},
   Year = {1997},
   Abstract = {We are concerned with the convergence of Lax-Weridroff type
             schemes with high resolution to the entropy solutions fo:
             conservation laws. These schemes include the original
             Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and
             its two step versions-the Richtrayer scheme and the
             MacCormack scheme. For the convex scalar conservation laws
             with algebraic growth flux functions, we prove the
             convergence of these schemes to the weak solutions
             satisfying appropriate entropy inequalities. The proof is
             based on detailed Lp estimates of the approximate solutions,
             H-1 compactness estimates of the corresponding entropy
             dissipation measures, and some compensated compactness
             frameworks. Then these techniques are generalized to study
             the convergence problem for the nonconvex scalar case and
             the hyperbolic systems of conservation laws.},
   Key = {fds246923}
}

@article{fds246924,
   Author = {Weinan, E and Liu, J-G},
   Title = {Finite difference schemes for incompressible flows in the
             velocity - impulse density formulation},
   Journal = {Journal of Computational Physics},
   Volume = {130},
   Number = {1},
   Pages = {67-76},
   Year = {1997},
   Abstract = {We consider finite difference schemes based on the impulse
             density variable. We show that the original velocity -
             impulse density formulation of Oseledets is marginally
             ill-posed for the inviscid flow, and this has the
             consequence that some ordinarily stable numerical methods in
             other formulations become unstable in the velocity - impulse
             density formulation. We present numerical evidence of this
             instability. We then discuss the construction of stable
             finite difference schemes by requiring that at the numerical
             level the nonlinear terms be convertible to similar terms in
             the primitive variable formulation. Finally we give a
             simplified velocity - impulse density formulation which is
             free of these complications and yet retains the nice
             features of the original velocity - impulse density
             formulation with regard to the treatment of boundary. We
             present numerical results on this simplified formulation for
             the driven cavity flow on both the staggered and
             non-staggered grids. © 1997 Academic Press.},
   Key = {fds246924}
}

@article{fds246914,
   Author = {Jin, S and Liu, J-G},
   Title = {Oscillations induced by numerical viscosities},
   Journal = {Mat. Contemp.},
   Volume = {10},
   Pages = {169-180},
   Year = {1996},
   Key = {fds246914}
}

@article{fds246915,
   Author = {Jin, S and Liu, J-G},
   Title = {The effects of numerical viscosities: I. Slowly moving
             shocks},
   Journal = {Journal of Computational Physics},
   Volume = {126},
   Number = {2},
   Pages = {373-389},
   Year = {1996},
   url = {http://dx.doi.org/10.1006/jcph.1996.0144},
   Abstract = {We begin a systematical study on the effect of numerical
             viscosities. In this paper we investigate the behavior of
             shock-capturing methods for slowly moving shocks. It is
             known that for slowly moving shocks even a first-order
             scheme, such as the Godunov or Roe type methods, will
             generate downstream oscillatory wave patterns that cannot be
             effectively damped by the dissipation of these first-order
             schemes. The purpose of this paper is to understand the
             formation and behavior of these downstream patterns. Our
             study shows that the downstream errors are generated by the
             unsteady nature of the viscous shock profiles and behave
             diffusively. The scenario is as follows. When solving the
             compressible Euler equations by shock capturing methods, the
             smeared density profile introduces a momentum spike at the
             shock location if the shock moves slowly. Downstream waves
             will necessarily emerge in order to balance the momentum
             mass carried by the spike for the momentum conservation.
             Although each family of waves decays in l∞ and l2 while
             they preserve the same mass, the perturbing nature of the
             viscous or spike profile is a constant source for the
             generation of new downstream waves, causing spurious
             solutions for all time. Higher order TVD or ENO type
             interpolations accentuate this problem. © 1996 Academic
             Press, Inc.},
   Doi = {10.1006/jcph.1996.0144},
   Key = {fds246915}
}

@article{fds246916,
   Author = {Weinan, E and Liu, J-G},
   Title = {Vorticity boundary condition and related issues for finite
             difference schemes},
   Journal = {Journal of Computational Physics},
   Volume = {124},
   Number = {2},
   Pages = {368-382},
   Year = {1996},
   url = {http://dx.doi.org/10.1006/jcph.1996.0066},
   Abstract = {This paper discusses three basic issues related to the
             design of finite difference schemes for unsteady viscous
             incompressible flows using vorticity formulations: the
             boundary condition for vorticity, an efficient time-stepping
             procedure, and the relation between these schemes and the
             ones based on velocity-pressure formulation. We show that
             many of the newly developed global vorticity boundary
             conditions can actually be written as some local formulas
             derived earlier. We also show that if we couple a standard
             centered difference scheme with third-or fourth-order
             explicit Runge-Kutta methods, the resulting schemes have no
             cell Reynolds number constraints. For high Reynolds number
             flows, these schemes are stable under the CFL condition
             given by the convective terms. Finally, we show that the
             classical MAC scheme is the same as Thom's formula coupled
             with second-order centered differences in the interior, in
             the sense that one can define discrete vorticity in a
             natural way for the MAC scheme and get the same values as
             the ones computed from Thom's formula. We use this to derive
             an efficient fourth-order Runge-Kutta time discretization
             for the MAC scheme from the one for Thom's formula. We
             present numerical results for driven cavity flow at high
             Reynolds number (105). © 1996 Academic Press,
             Inc.},
   Doi = {10.1006/jcph.1996.0066},
   Key = {fds246916}
}

@article{fds246917,
   Author = {Weinan, E and Liu, J-G},
   Title = {Essentially compact schemes for unsteady viscous
             incompressible flows},
   Journal = {Journal of Computational Physics},
   Volume = {126},
   Number = {1},
   Pages = {122-138},
   Year = {1996},
   url = {http://dx.doi.org/10.1006/jcph.1996.0125},
   Abstract = {A new fourth-order accurate finite difference scheme for the
             computation of unsteady viscous incompressible flows is
             introduced. The scheme is based on the vorticity-stream
             function formulation. It is essentially compact and has the
             nice features of a compact scheme with regard to the
             treatment of boundary conditions. It is also very efficient,
             at every time step or Runge-Kutta stage, only two
             Poisson-like equations have to be solved. The Poisson-like
             equations are amenable to standard fast Poisson solvers
             usually designed for second order schemes. Detailed
             comparison with the second-order scheme shows the clear
             superiority of this new fourth-order scheme in resolving
             both the boundary layers and the gross features of the flow.
             This efficient fourth-order scheme also made it possible to
             compute the driven cavity flow at Reynolds number 106 on a
             10242 grid at a reasonable cost. Fourth-order convergence is
             proved under mild regularity requirements. This is the first
             such result to our knowledge. © 1996 Academic Press,
             Inc.},
   Doi = {10.1006/jcph.1996.0125},
   Key = {fds246917}
}

@article{fds246918,
   Author = {Weinan, E and Liu, J-G},
   Title = {Projection method II: Godunov-Ryabenki analysis},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {33},
   Number = {4},
   Pages = {1597-1621},
   Year = {1996},
   Abstract = {This is the second of a series of papers on the subject of
             projection methods for viscous incompressible flow
             calculations. The purpose of the present paper is to explain
             why the accuracy of the velocity approximation is not
             affected by (1) the numerical boundary layers in the
             approximation of pressure and the intermediate velocity
             field and (2) the noncommutativity of the projection
             operator and the laplacian. This is done by using a
             Godunov-Ryabenki type of analysis in a rigorous fashion. By
             doing so, we hope to be able to convey the message that
             normal mode analysis is basically sufficient for
             understanding the stability and accuracy of a
             finite-difference method for the Navier-Stokes equation even
             in the presence of boundaries. As an example, we analyze the
             second-order projection method based on pressure increment
             formulations used by van Kan and Bell, Colella, and Glaz.
             The leading order error term in this case is of O(Δt) and
             behaves as high frequency oscillations over the whole
             domain, compared with the O(Δt1/2) numerical boundary
             layers found in the second-order Kim-Moin
             method.},
   Key = {fds246918}
}

@article{fds246919,
   Author = {Levermore, CD and Liu, J-G},
   Title = {Large oscillations arising in a dispersive numerical
             scheme},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {99},
   Number = {2-3},
   Pages = {191-216},
   Year = {1996},
   Abstract = {We study the oscillatory behavior that arises in solutions
             of a dispersive numerical scheme for the Hopf equation
             whenever the classical solution of that equation develops a
             singularity. Modulation equations are derived that describe
             period-two oscillations so long as the solution of those
             equations takes values for which the equations are
             hyperbolic. However, those equations have an elliptic region
             that may be entered by its solutions in a unite time, after
             which the corresponding period-two oscillations are seen to
             break down. This kind of phenomenon has not been observed
             for integrable schemes. The generation and propagation of
             period-two oscillations are asymptotically analyzed and a
             matching formula is found for the transition between
             oscillatory and nonoscillatory regions. Modulation equations
             are also presented for period-three oscillations. Numerical
             experiments are carried out that illustrate our analysis. ©
             1996 Elsevier Science B.V. All rights reserved.},
   Key = {fds246919}
}

@article{fds246920,
   Author = {Liu, J-G and Xin, Z},
   Title = {Kinetic and viscous boundary layers for broadwell
             equations},
   Journal = {Transport Theory and Statistical Physics},
   Volume = {25},
   Number = {3-5},
   Pages = {447-461},
   Year = {1996},
   Abstract = {In this paper, we investigate the boundary layer behavior of
             solutions to the one dimensional Broadwell model of the
             nonlinear Boltzmann equation for small mean free path. We
             consider the analogue of Maxwell's diffusive and the
             reflexive boundary conditions. It is found that even for
             such a simple model, there are boundary layers due to purely
             kinetic effects which cannot be detected by the
             corresponding Navier-Stokes system. It is also found
             numerically that a compressive boundary layer is not always
             stable in the sense that it may detach from the boundary and
             move into the interior of the gas as a shock
             layer.},
   Key = {fds246920}
}

@article{fds246921,
   Author = {Liu, J-G and Xin, Z},
   Title = {Boundary layer behavior in the fluid-dynamic limit for a
             nonlinear model Boltzmann equation},
   Journal = {Arch. Rat. Mech. Anal.},
   Volume = {135},
   Pages = {61-105},
   Year = {1996},
   Key = {fds246921}
}

@article{fds246912,
   Author = {Weinan, E and Liu, J-G},
   Title = {Projection method I: convergence and numerical boundary
             layers},
   Journal = {SIAM J. Numer. Anal.},
   Volume = {32},
   Pages = {1017-1057},
   Year = {1995},
   Key = {fds246912}
}

@article{fds246913,
   Author = {Liu, J-G and Xin, Z},
   Title = {Convergence of vortex methods for weak solutions to the 2-D
             Euler equations with vortex sheets data},
   Journal = {Comm. Pure Appl. Math.},
   Volume = {48},
   Pages = {611-628},
   Year = {1995},
   Key = {fds246913}
}

@article{fds246910,
   Author = {Lefloch, P and Liu, J-G},
   Title = {Discrete entropy and monotonicity criteria for hyperbolic
             conservation laws},
   Journal = {C.R. Acad. Sci. Paris.},
   Volume = {319},
   Pages = {881-886},
   Year = {1994},
   Key = {fds246910}
}

@article{fds246911,
   Author = {Jin, S and Liu, J-G},
   Title = {Relaxation and diffusion enhanced dispersive
             waves},
   Journal = {Proceedings of The Royal Society of London, Series A:
             Mathematical and Physical Sciences},
   Volume = {446},
   Number = {1928},
   Pages = {555-563},
   Year = {1994},
   Abstract = {The development of shocks in nonlinear hyperbolic
             conservation laws may be regularized through either
             diffusion or relaxation. However, we have observed
             surprisingly that for some physical problems, when both of
             the smoothing factors diffusion and relaxation coexist,
             under appropriate asymptotic assumptions, the dispersive
             waves are enhanced. This phenomenon is studied
             asymptotically in the sense of the Chapman-Enskog expansion
             and demonstrated numerically.},
   Key = {fds246911}
}

@article{fds246906,
   Author = {Chen, G-Q and Liu, J-G},
   Title = {Convergence of second-order schemes for isentropic gas
             dynamics},
   Journal = {Math. Comp.},
   Volume = {61},
   Pages = {607-629},
   Year = {1993},
   Key = {fds246906}
}

@article{fds246907,
   Author = {Engquist, B and Liu, J-G},
   Title = {Numerical methods for oscillatory solutions to hyperbolic
             problems},
   Journal = {Comm. Pure Appl. Math.},
   Volume = {46},
   Pages = {1327-1361},
   Year = {1993},
   Key = {fds246907}
}

@article{fds246908,
   Author = {Liu, J-G and Xin, Z},
   Title = {L1-stability of stationary discrete shocks},
   Journal = {Math. Comp.},
   Volume = {60},
   Pages = {233-244},
   Year = {1993},
   Key = {fds246908}
}

@article{fds246909,
   Author = {Liu, J-G and Xin, Z},
   Title = {Nonlinear stability of discrete shocks for systems of
             conservation laws},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {125},
   Number = {3},
   Pages = {217-256},
   Year = {1993},
   ISSN = {0003-9527},
   url = {http://dx.doi.org/10.1007/BF00383220},
   Abstract = {In this paper we study the asymptotic nonlinear stability of
             discrete shocks for the Lax-Friedrichs scheme for
             approximating general m×m systems of nonlinear hyperbolic
             conservation laws. It is shown that weak single discrete
             shocks for such a scheme are nonlinearly stable in the
             Lp-norm for all p ≧ 1, provided that the sums of the
             initial perturbations equal zero. These results should shed
             light on the convergence of the numerical solution
             constructed by the Lax-Friedrichs scheme for the
             single-shock solution of system of hyperbolic conservation
             laws. If the Riemann solution corresponding to the given
             far-field states is a superposition of m single shocks from
             each characteristic family, we show that the corresponding
             multiple discrete shocks are nonlinearly stable in Lp (P ≧
             2). These results are proved by using both a weighted
             estimate and a characteristic energy method based on the
             internal structures of the discrete shocks and the essential
             monotonicity of the Lax-Friedrichs scheme. © 1993
             Springer-Verlag.},
   Doi = {10.1007/BF00383220},
   Key = {fds246909}
}


%% Papers Accepted   
@article{fds325700,
   Author = {Degond, P and Liu, J-G and Pego, RL},
   Title = {Coagulation–Fragmentation Model for Animal Group-Size
             Statistics},
   Journal = {Journal of Nonlinear Science},
   Volume = {27},
   Number = {2},
   Pages = {379-424},
   Year = {2017},
   Month = {April},
   url = {http://dx.doi.org/10.1007/s00332-016-9336-3},
   Doi = {10.1007/s00332-016-9336-3},
   Key = {fds325700}
}

@article{fds327636,
   Author = {Huang, H and Liu, J-G},
   Title = {Error estimate of a random particle blob method for the
             Keller-Segel equation},
   Journal = {Mathematics of Computation},
   Volume = {86},
   Number = {308},
   Pages = {2719-2744},
   Year = {2017},
   Month = {February},
   url = {http://dx.doi.org/10.1090/mcom/3174},
   Doi = {10.1090/mcom/3174},
   Key = {fds327636}
}

@article{fds325701,
   Author = {Liu, J-G and Wang, J},
   Title = {Global existence for a thin film equation with subcritical
             mass},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {22},
   Number = {4},
   Pages = {1461-1492},
   Year = {2017},
   Month = {February},
   url = {http://dx.doi.org/10.3934/dcdsb.2017070},
   Doi = {10.3934/dcdsb.2017070},
   Key = {fds325701}
}

@article{fds320739,
   Author = {P. Degond and J.-G. Liu and S. Merino-Aceituno and T.
             Tardiveau},
   Title = {Continuum dynamics of the intention field under weakly
             cohesive social interactions},
   Journal = {Math. Models Methods Appl. Sci.},
   Year = {2016},
   Key = {fds320739}
}

@article{fds320743,
   Author = {Y. Gao and J.-G. Liu and J. Lu},
   Title = {Continuum limit of a mesoscopic model of step motion on
             vicinal surfaces},
   Journal = {J. Nonlinear Science},
   Year = {2016},
   Key = {fds320743}
}

@article{fds300227,
   Author = {J.-G. Liu and R. Yang},
   Title = {A random particle blob method for the Keller-Segel equation
             and convergence analysis},
   Journal = {Math. Comp.},
   Year = {2015},
   Key = {fds300227}
}