## 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{fds333565,
Author = {Liu, J-G and Xu, X},
Title = {Partial regularity of weak solutions to a PDE system with
cubic nonlinearity},
Journal = {Journal of Differential Equations},
Volume = {264},
Number = {8},
Pages = {5489-5526},
Year = {2018},
Month = {April},
url = {http://dx.doi.org/10.1016/j.jde.2018.01.0010022},
Doi = {10.1016/j.jde.2018.01.0010022},
Key = {fds333565}
}

@article{fds333566,
Author = {Li, L and Liu, JG},
Title = {p-Euler equations and p-Navier-Stokes equations},
Journal = {Journal of Differential Equations},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1016/j.jde.2017.12.023},
Abstract = {© 2017 Elsevier Inc. We propose in this work new systems of
equations which we call p-Euler equations and
p-Navier-Stokes equations. p-Euler equations are derived as
the Euler-Lagrange equations for the action represented by
the Benamou-Brenier characterization of Wasserstein-p
distances, with incompressibility constraint. p-Euler
equations have similar structures with the usual Euler
equations but the 'momentum' is the signed (p-1)-th power of
the velocity. In the 2D case, the p-Euler equations have
streamfunction-vorticity formulation, where the vorticity is
given by the p-Laplacian of the streamfunction. By adding
diffusion presented by γ-Laplacian of the velocity, we
obtain what we call p-Navier-Stokes equations. If γ=p, the
a priori energy estimates for the velocity and momentum have
dual symmetries. Using these energy estimates and a
time-shift estimate, we show the global existence of weak
solutions for the p-Navier-Stokes equations in Rd for γ=p
and p≥d≥2 through a compactness criterion.},
Doi = {10.1016/j.jde.2017.12.023},
Key = {fds333566}
}

@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{fds331396,
Author = {Liu, J-G and Wang, J},
Title = {A generalized Sz. Nagy inequality in higher dimensions and
the critical thin film equation},
Journal = {Nonlinearity},
Volume = {30},
Number = {1},
Pages = {35-60},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1088/0951-7715/30/1/35},
Doi = {10.1088/0951-7715/30/1/35},
Key = {fds331396}
}

@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{fds330536,
Author = {Liu, J-G and Xu, X},
Title = {Analytical Validation of a Continuum Model for the Evolution
of a Crystal Surface in Multiple Space Dimensions},
Journal = {SIAM Journal on Mathematical Analysis},
Volume = {49},
Number = {3},
Pages = {2220-2245},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1137/16M1098474},
Doi = {10.1137/16M1098474},
Key = {fds330536}
}

@article{fds329523,
Author = {Huang, H and Liu, J-G},
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},
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{fds330537,
Author = {Degond, P and Herty, M and Liu, J-G},
Title = {Mean-field games and model predictive control},
Journal = {Communications in Mathematical Sciences},
Volume = {15},
Number = {5},
Pages = {1403-1422},
Year = {2017},
url = {http://dx.doi.org/10.4310/CMS.2017.v15.n5.a9},
Doi = {10.4310/CMS.2017.v15.n5.a9},
Key = {fds330537}
}

@article{fds333567,
Author = {Li, L and Liu, J-G},
Title = {A note on deconvolution with completely monotone sequences
and discrete fractional calculus},
Journal = {Quarterly of Applied Mathematics},
Pages = {1-1},
Year = {2017},
url = {http://dx.doi.org/10.1090/qam/1479},
Doi = {10.1090/qam/1479},
Key = {fds333567}
}

@article{fds333568,
Author = {Coquel, F and Jin, S and Liu, J-G and Wang, L},
Title = {Entropic sub-cell shock capturing schemes via Jin-Xin
relaxation and Glimm front sampling for scalar conservation
laws},
Journal = {Mathematics of Computation},
Pages = {1-1},
Year = {2017},
url = {http://dx.doi.org/10.1090/mcom/3253},
Doi = {10.1090/mcom/3253},
Key = {fds333568}
}

@article{fds333569,
Author = {Liu, J-G and Wang, L and Zhou, Z},
Title = {Positivity-preserving and asymptotic preserving method for
2D Keller-Segal equations},
Journal = {Mathematics of Computation},
Pages = {1-1},
Year = {2017},
url = {http://dx.doi.org/10.1090/mcom/3250},
Doi = {10.1090/mcom/3250},
Key = {fds333569}
}

@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{fds333570,
Author = {Liu, J-G and Wang, J},
Title = {Refined hyper-contractivity and uniqueness for the
Keller–Segel equations},
Journal = {Applied Mathematics Letters},
Volume = {52},
Pages = {212-219},
Year = {2016},
Month = {February},
url = {http://dx.doi.org/10.1016/j.aml.2015.09.001},
Doi = {10.1016/j.aml.2015.09.001},
Key = {fds333570}
}

@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{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.
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{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},
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{fds333571,
Author = {Degond, P and Frouvelle, A and Liu, J-G},
Title = {A NOTE ON PHASE TRANSITIONS FOR THE SMOLUCHOWSKI EQUATION
WITH DIPOLAR POTENTIAL},
Journal = {HYPERBOLIC PROBLEMS: THEORY, NUMERICS, APPLICATIONS},
Volume = {8},
Pages = {179-192},
Booktitle = {Proceedings of the Fourteenth International Conference on
Hyperbolic Problems: Theory, Numerics and
Application},
Year = {2014},
Key = {fds333571}
}

@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
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
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
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. ©
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 &lt;
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 &gt; 1 - 2/d,
the weak solution preserves mass and satisfies the
hyper-contractive estimates in L q for any p &lt; q &lt;
∞. Furthermore, for slow diffusion 1 &lt; 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 &gt; 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
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
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},
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&gt;N, and local existence of unique classical solutions
in H k(ℝ N),k&gt; 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 &gt; N/2 + 3. For the
stationary weak solutions of the Euler-Poincaré equations
we prove a Liouville type theorem. Namely, for α &gt; 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
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 &lt; 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,
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
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
New York.},
Doi = {10.1007/s00332-012-9157-y},
Key = {fds246895}
}

@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.
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.
Key = {fds304584}
}

@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,
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.
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
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
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{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
Inc.},
Doi = {10.1002/cpa.20178},
Key = {fds246880}
}

@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{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{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
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
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 |Î” h| 2, which represents the
surface diffusion, and-log (1 + |âˆ‡ 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 Îµ 2 of |Î” h| 2 with Îµ = 1/L and L the
linear size of the underlying system, and study the energy
asymptotics in the large system limit Îµ â†’ 0. We
show that global minimizers of the free-energy functional
exist for each Îµ &gt; 0, the L 2-norm of the gradient of
any global minimizer scales as O(1/Îµ), and the global
minimum energy scales as O( log Îµ). 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
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},
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
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,
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. Â©
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
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 &gt; 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
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 &amp; 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 &amp; 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
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
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
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
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
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
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
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
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
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. ©
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{fds332012,
Author = {Liu, J-G and Yang, R},
Title = {A random particle blob method for the Keller-Segel equation
and convergence analysis},
Journal = {Mathematics of Computation},
Volume = {86},
Number = {304},
Pages = {725-745},
Year = {2016},
Month = {May},
url = {http://dx.doi.org/10.1090/mcom/3118},
Doi = {10.1090/mcom/3118},
Key = {fds332012}
}

@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}
}