%% 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{fds335603, Author = {Feng, Y and Li, L and Liu, J-G and Xu, X}, Title = {A note on one-dimensional time fractional ODEs}, Journal = {Applied Mathematics Letters}, Volume = {83}, Pages = {87-94}, Year = {2018}, Month = {September}, url = {http://dx.doi.org/10.1016/j.aml.2018.03.015}, Doi = {10.1016/j.aml.2018.03.015}, Key = {fds335603} } @article{fds335604, Author = {Li, L and Liu, J-G and Wang, L}, Title = {Cauchy problems for Keller–Segel type time–space fractional diffusion equation}, Journal = {Journal of Differential Equations}, Volume = {265}, Number = {3}, Pages = {1044-1096}, Year = {2018}, Month = {August}, url = {http://dx.doi.org/10.1016/j.jde.2018.03.025}, Abstract = {© 2018 Elsevier Inc. This paper investigates Cauchy problems for nonlinear fractional time–space generalized Keller–Segel equation Dtβ0cρ+(−△)[Formula presented]ρ+∇⋅(ρB(ρ))=0, where Caputo derivative Dtβ0cρ models memory effects in time, fractional Laplacian (−△)[Formula presented]ρ represents Lévy diffusion and B(ρ)=−sn,γ∫Rn[Formula presented]ρ(y)dy is the Riesz potential with a singular kernel which takes into account the long rang interaction. We first establish Lr−Lqestimates and weighted estimates of the fundamental solutions (P(x,t),Y(x,t)) (or equivalently, the solution operators (Sαβ(t),Tαβ(t))). Then, we prove the existence and uniqueness of the mild solutions when initial data are in Lpspaces, or the weighted spaces. Similar to Keller–Segel equations, if the initial data are small in critical space Lpc(Rn) (pc=[Formula presented]), we construct the global existence. Furthermore, we prove the L1integrability and integral preservation when the initial data are in L1(Rn)∩Lp(Rn) or L1(Rn)∩Lpc(Rn). Finally, some important properties of the mild solutions including the nonnegativity preservation, mass conservation and blowup behaviors are established.}, Doi = {10.1016/j.jde.2018.03.025}, Key = {fds335604} } @article{fds335605, Author = {Liu, JG and Tang, M and Wang, L and Zhou, Z}, Title = {An accurate front capturing scheme for tumor growth models with a free boundary limit}, Journal = {Journal of Computational Physics}, Volume = {364}, Pages = {73-94}, Year = {2018}, Month = {July}, url = {http://dx.doi.org/10.1016/j.jcp.2018.03.013}, Abstract = {© 2018 Elsevier Inc. We consider a class of tumor growth models under the combined effects of density-dependent pressure and cell multiplication, with a free boundary model as its singular limit when the pressure-density relationship becomes highly nonlinear. In particular, the constitutive law connecting pressure p and density ρ is p(ρ)=[Formula presented]ρm−1, and when m≫1, the cell density ρ may evolve its support according to a pressure-driven geometric motion with sharp interface along its boundary. The nonlinearity and degeneracy in the diffusion bring great challenges in numerical simulations. Prior to the present paper, there is lack of standard mechanism to numerically capture the front propagation speed as m≫1. In this paper, we develop a numerical scheme based on a novel prediction-correction reformulation that can accurately approximate the front propagation even when the nonlinearity is extremely strong. We show that the semi-discrete scheme naturally connects to the free boundary limit equation as m→∞. With proper spatial discretization, the fully discrete scheme has improved stability, preserves positivity, and can be implemented without nonlinear solvers. Finally, extensive numerical examples in both one and two dimensions are provided to verify the claimed properties in various applications.}, Doi = {10.1016/j.jcp.2018.03.013}, Key = {fds335605} } @article{fds335606, Author = {Chen, K and Li, Q and Liu, J-G}, Title = {Online learning in optical tomography: a stochastic approach}, Journal = {Inverse Problems}, Volume = {34}, Number = {7}, Pages = {075010-075010}, Year = {2018}, Month = {July}, url = {http://dx.doi.org/10.1088/1361-6420/aac220}, Doi = {10.1088/1361-6420/aac220}, Key = {fds335606} } @article{fds335607, Author = {Gao, Y and Liu, J-G and Lu, XY and Xu, X}, Title = {Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface}, Journal = {Calculus of Variations and Partial Differential Equations}, Volume = {57}, Number = {2}, Year = {2018}, Month = {April}, url = {http://dx.doi.org/10.1007/s00526-018-1326-x}, Doi = {10.1007/s00526-018-1326-x}, Key = {fds335607} } @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.2018.01.001}, Abstract = {© 2018 Elsevier Inc. In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.}, Doi = {10.1016/j.jde.2018.01.001}, Key = {fds333565} } @article{fds333566, Author = {Li, L and Liu, J-G}, Title = {p -Euler equations and p -Navier–Stokes equations}, Journal = {Journal of Differential Equations}, Volume = {264}, Number = {7}, Pages = {4707-4748}, Year = {2018}, Month = {April}, 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{fds338622, Author = {Feng, Y and Li, L and Liu, JG}, Title = {Semigroups of stochastic gradient descent and online principal component analysis: Properties and diffusion approximations}, Journal = {Communications in Mathematical Sciences}, Volume = {16}, Number = {3}, Pages = {777-789}, Year = {2018}, Month = {January}, url = {http://dx.doi.org/10.4310/CMS.2018.v16.n3.a7}, Abstract = {© 2018 International Press. We study the Markov semigroups for two important algorithms from machine learning: stochastic gradient descent (SGD) and online principal component analysis (PCA). We investigate the effects of small jumps on the properties of the semigroups. Properties including regularity preserving, L∞contraction are discussed. These semigroups are the dual of the semigroups for evolution of probability, while the latter are L1contracting and positivity preserving. Using these properties, we show that stochastic differential equations (SDEs) in Rd (on the sphere Sd-1) can be used to approximate SGD (online PCA) weakly. These SDEs may be used to provide some insights of the behaviors of these algorithms.}, Doi = {10.4310/CMS.2018.v16.n3.a7}, Key = {fds338622} } @article{fds338623, Author = {Li, L and Liu, J-G}, Title = {SOME COMPACTNESS CRITERIA FOR WEAK SOLUTIONS OF TIME FRACTIONAL PDEs}, Journal = {Siam Journal on Mathematical Analysis}, Volume = {50}, Number = {4}, Pages = {3963-3995}, Publisher = {SIAM PUBLICATIONS}, Year = {2018}, Month = {January}, url = {http://dx.doi.org/10.1137/17M1145549}, Doi = {10.1137/17M1145549}, Key = {fds338623} } @article{fds335608, Author = {Gao, Y and Li, L and Liu, J-G}, Title = {A Dispersive Regularization for the Modified Camassa--Holm Equation}, Journal = {Siam Journal on Mathematical Analysis}, Volume = {50}, Number = {3}, Pages = {2807-2838}, Year = {2018}, Month = {January}, url = {http://dx.doi.org/10.1137/17M1132756}, Doi = {10.1137/17M1132756}, Key = {fds335608} } @article{fds335609, Author = {Li, L and Liu, J-G}, Title = {A Generalized Definition of Caputo Derivatives and Its Application to Fractional ODEs}, Journal = {Siam Journal on Mathematical Analysis}, Volume = {50}, Number = {3}, Pages = {2867-2900}, Year = {2018}, Month = {January}, url = {http://dx.doi.org/10.1137/17M1160318}, Doi = {10.1137/17M1160318}, Key = {fds335609} } @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{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}, Volume = {87}, Number = {311}, Pages = {1165-1189}, Year = {2017}, Month = {September}, url = {http://dx.doi.org/10.1090/mcom/3250}, Doi = {10.1090/mcom/3250}, Key = {fds333569} } @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}, Volume = {87}, Number = {311}, Pages = {1083-1126}, Year = {2017}, Month = {September}, url = {http://dx.doi.org/10.1090/mcom/3253}, Doi = {10.1090/mcom/3253}, Key = {fds333568} } @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{fds338528, Author = {Gao, Y and Ji, H and Liu, J-G and P. Witelski and T}, Title = {A vicinal surface model for epitaxial growth with logarithmic free energy}, Journal = {Discrete & Continuous Dynamical Systems B}, Volume = {22}, Number = {11}, Pages = {1-21}, Year = {2017}, url = {http://dx.doi.org/10.3934/dcdsb.2018170}, Doi = {10.3934/dcdsb.2018170}, Key = {fds338528} } @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{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. Our main assumptions are: adsorbed atoms (adatoms) form a dilute system, and elastic effects of the crystal lattice are absent. The step edge is treated as a front that propagates via probabilistic rules for atom attachment and detachment at the step. We formally derive a quasistatic step flow description by averaging out the stochastic scheme when terrace diffusion, adatom desorption, and deposition from above are present.}, Doi = {10.1103/physreve.91.032403}, Key = {fds246843} } @article{fds300223, Author = {Degond, P and Frouvelle, A and Liu, JG}, Title = {Phase Transitions, Hysteresis, and Hyperbolicity for Self-Organized Alignment Dynamics}, Journal = {Archive for Rational Mechanics and Analysis}, Volume = {216}, Number = {1}, Pages = {63-115}, Year = {2015}, Month = {January}, ISSN = {0003-9527}, url = {http://dx.doi.org/10.1007/s00205-014-0800-7}, Abstract = {© 2014, Springer-Verlag Berlin Heidelberg. We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function.}, Doi = {10.1007/s00205-014-0800-7}, Key = {fds300223} } @article{fds300222, Author = {Chertock, A and Liu, JG and Pendleton, T}, Title = {Elastic collisions among peakon solutions for the Camassa-Holm equation}, Journal = {Applied Numerical Mathematics}, Volume = {93}, Pages = {30-46}, Year = {2015}, Month = {January}, ISSN = {0168-9274}, url = {http://dx.doi.org/10.1016/j.apnum.2014.01.001}, Abstract = {© 2014 IMACS. The purpose of this paper is to study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa-Holm equation. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a particle method to the Camassa-Holm equation and show that the nonlinear interaction among the peakon solutions resembles an elastic collision, i.e., the total energy and momentum of the system before the peakon interaction is equal to the total energy and momentum of the system after the collision. From this result, we provide several numerical illustrations which support the analytical study, as well as showcase the merits of using a particle method to simulate solutions to the Camassa-Holm equation under a wide class of initial data.}, Doi = {10.1016/j.apnum.2014.01.001}, Key = {fds300222} } @article{fds313338, Author = {Herschlag, G and Liu, J-G and Layton, AT}, Title = {An Exact Solution for Stokes Flow in a Channel with Arbitrarily Large Wall Permeability}, Journal = {Siam Journal on Applied Mathematics}, Volume = {75}, Number = {5}, Pages = {2246-2267}, Year = {2015}, Month = {January}, ISSN = {0036-1399}, url = {http://dx.doi.org/10.1137/140995854}, Doi = {10.1137/140995854}, Key = {fds313338} } @article{fds246846, Author = {Degond, P and Liu, J-G and Ringhofer, C}, Title = {Evolution of wealth in a non-conservative economy driven by local Nash equilibria}, Journal = {Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences}, 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{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) Non Linear Analysis}, Volume = {31}, Number = {3}, Pages = {555-565}, Year = {2014}, Month = {May}, 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{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 Series A}, 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{fds337236, Author = {Chae, D and Degond, P and Liu, JG}, Title = {Well-posedness for hall-magnetohydrodynamics}, Journal = {Annales De L'Institut Henri Poincare (C) Non Linear Analysis}, Volume = {31}, Number = {3}, Pages = {555-565}, Year = {2014}, Month = {January}, 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 = {fds337236} } @article{fds246851, Author = {Chen, X and Li, X and Liu, J-G}, Title = {Existence and uniqueness of global weak solution to a kinetic model for the sedimentation of rod-like particles}, Journal = {Communications in Mathematical Sciences}, Volume = {12}, Number = {8}, Pages = {1579-1601}, Year = {2014}, ISSN = {1539-6746}, url = {http://dx.doi.org/10.4310/CMS.2014.v12.n8.a10}, Doi = {10.4310/CMS.2014.v12.n8.a10}, Key = {fds246851} } @article{fds246857, Author = {Johnston, H and Wang, C and Liu, J-G}, Title = {A Local Pressure Boundary Condition Spectral Collocation Scheme for the Three-Dimensional Navier–Stokes Equations}, Journal = {Journal of Scientific Computing}, Volume = {60}, Number = {3}, Pages = {612-626}, Year = {2014}, ISSN = {0885-7474}, url = {http://dx.doi.org/10.1007/s10915-013-9808-7}, Abstract = {© 2014, Springer Science+Business Media New York.A spectral collocation scheme for the three-dimensional incompressible (u,p) formulation of the Navier–Stokes equations, in domains Ω with a non-periodic boundary condition, is described. The key feature is the high order approximation, by means of a local Hermite interpolant, of a Neumann boundary condition for use in the numerical solution of the pressure Poisson system. The time updates of the velocity u and pressure p are decoupled as a result of treating the pressure gradient in the momentum equation explicitly in time. The pressure update is computed from a pressure Poisson equation. Extension of the overall methodology to the Boussinesq system is also described. The uncoupling of the pressure and velocity time updates results in a highly efficient scheme that is simple to implement and well suited for simulating moderate to high Reynolds and Rayleigh number flows. Accuracy checks are presented, along with simulations of the lid-driven cavity flow and a differentially heated cavity flow, to demonstrate the scheme produces accurate three-dimensional results at a reasonable computational cost.}, Doi = {10.1007/s10915-013-9808-7}, Key = {fds246857} } @article{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 efficiency of our scheme. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.}, Doi = {10.1137/120875508}, Key = {fds246861} } @article{fds220117, Author = {P. Degond and A. Frouvelle and J.-G. Liu and S Motsch and L Navoret}, Title = {Macroscopic models of collective motion and self-organization}, Journal = {Seminaire Laurent Schwartz -- EDP et applicatios}, Volume = {2012 - 2013}, Pages = {1-27}, Year = {2013}, Key = {fds220117} } @article{fds220119, Author = {P. Degond and J.-G, Liu and S. Motsch and V. Panferov}, Title = {Hydrodynamic models of self-organized dynamics: derivation and existence theory}, Journal = {Methods Anal. Appl.}, Volume = {20}, Pages = {89-114}, Year = {2013}, Key = {fds220119} } @article{fds246859, Author = {Degond, P and Liu, J-G and Ringhofer, C}, Title = {Evolution of the Distribution of Wealth in an Economic Environment Driven by Local Nash Equilibria}, Journal = {Journal of Statistical Physics}, Volume = {154}, Number = {3}, Pages = {1-30}, Year = {2013}, ISSN = {0022-4715}, url = {http://dx.doi.org/10.1007/s10955-013-0888-4}, Abstract = {We present and analyze a model for the evolution of the wealth distribution within a heterogeneous economic environment. The model considers a system of rational agents interacting in a game theoretical framework, through fairly general assumptions on the cost function. This evolution drives the dynamic of the agents in both wealth and economic configuration variables. We consider a regime of scale separation where the large scale dynamics is given by a hydrodynamic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse gamma distribution as an equilibrium in the particular case of quadratic cost functions which has been previously considered in the literature. © 2013 Springer Science+Business Media New York.}, Doi = {10.1007/s10955-013-0888-4}, Key = {fds246859} } @article{fds246860, Author = {Chen, X and Jüngel, A and Liu, J-G}, Title = {A Note on Aubin-Lions-Dubinskiǐ Lemmas}, Journal = {Acta Applicandae Mathematicae}, Volume = {133}, Number = {1}, Pages = {1-11}, Year = {2013}, ISSN = {0167-8019}, url = {http://dx.doi.org/10.1007/s10440-013-9858-8}, Abstract = {Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the Aubin-Lions-Dubinskiǐ lemma are proven, refining some recent results in the literature. The first theorem sharpens slightly a result of Dubinskiǐ (in Mat. Sb. 67(109):609-642, 1965) for seminormed cones. The second theorem applies to piecewise constant functions in time and sharpens slightly the results of Dreher and Jüngel (in Nonlinear Anal. 75:3072-3077, 2012) and Chen and Liu (in Appl. Math. Lett. 25:2252-2257, 2012). An application is given, which is useful in the study of porous-medium or fast-diffusion type equations. © 2013 Springer Science+Business Media.}, Doi = {10.1007/s10440-013-9858-8}, Key = {fds246860} } @article{fds246863, Author = {Degond, P and Liu, J-G and Ringhofer, C}, Title = {Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria}, Journal = {Journal of Nonlinear Science}, Volume = {24}, Number = {1}, Pages = {1-23}, Year = {2013}, ISSN = {0938-8974}, url = {http://dx.doi.org/10.1007/s00332-013-9185-2}, Abstract = {We introduce a new mean field kinetic model for systems of rational agents interacting in a game-theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of the system is given by a macroscopic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. An application of the presented theory to a social model (herding behavior) is discussed. © 2013 Springer Science+Business Media New York.}, Doi = {10.1007/s00332-013-9185-2}, Key = {fds246863} } @article{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 = {1-54}, Year = {2013}, ISSN = {0010-3616}, url = {http://dx.doi.org/10.1007/s00220-013-1777-z}, Abstract = {This paper investigates infinite-time spreading and finite-time blow-up for the Keller-Segel system. For 0 < m ≤ 2 - 2 / d, the Lp space for both dynamic and steady solutions are detected with {Mathematical expression} . Firstly, the global existence of the weak solution is proved for small initial data in Lp. Moreover, when m > 1 - 2 / d, the weak solution preserves mass and satisfies the hyper-contractive estimates in Lq for any p < q < ∞. Furthermore, for slow diffusion 1 < m ≤ 2 - 2/d, this weak solution is also a weak entropy solution which blows up at finite time provided by the initial negative free energy. For m > 2 - 2/d, the hyper-contractive estimates are also obtained. Finally, we focus on the Lp norm of the steady solutions, it is shown that the energy critical exponent m = 2 d/(d + 2) is the critical exponent separating finite Lp norm and infinite Lp norm for the steady state solutions. © 2013 Springer-Verlag Berlin Heidelberg.}, Doi = {10.1007/s00220-013-1777-z}, Key = {fds246866} } @article{fds246869, Author = {Goudon, T and Jin, S and Liu, J-G and Yan, B}, Title = {Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows}, Journal = {Journal of Computational Physics}, Volume = {246}, Pages = {145-164}, Year = {2013}, ISSN = {0021-9991}, url = {http://dx.doi.org/10.1016/j.jcp.2013.03.038}, Abstract = {We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme is asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has a numerical stability condition controlled by the non-stiff convection operator, with an implicit treatment of the stiff drag term and the Fokker-Planck operator. Yet, consistent to a standard asymptotic-preserving Fokker-Planck solver or an incompressible Navier-Stokes solver, only the conjugate-gradient method and fast Poisson and Helmholtz solvers are needed. Numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the scheme, with several interesting applications. © 2013 Elsevier Inc.}, Doi = {10.1016/j.jcp.2013.03.038}, Key = {fds246869} } @article{fds246870, Author = {Chen, X and Liu, J-G}, Title = {Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions}, Journal = {Journal of Differential Equations}, Volume = {254}, Number = {7}, Pages = {2764-2802}, Year = {2013}, ISSN = {0022-0396}, url = {http://dx.doi.org/10.1016/j.jde.2013.01.005}, Abstract = {We prove the existence of the global weak entropy solution to the Doi-Saintillan-Shelley model for active and passive rod-like particle suspensions, which couples a Fokker-Planck equation with the incompressible Navier-Stokes or Stokes equation, under the no-flux boundary conditions, L2(Ω;L1(Sd-1)) initial data, and finite initial entropy for the particle distribution function in two and three dimensions. Furthermore, for the model with the Stokes equation, we obtain the global L2(Ω×Sd-1) weak solution in two and three dimensions and the uniqueness in two dimension. © 2013 Elsevier Inc..}, Doi = {10.1016/j.jde.2013.01.005}, Key = {fds246870} } @article{fds246894, Author = {Haack, J and Jin, S and Liu, J}, Title = {An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations}, Journal = {Communications in Computational Physics}, Volume = {12}, Number = {04}, Pages = {955-980}, Year = {2012}, Month = {October}, ISSN = {1815-2406}, url = {http://dx.doi.org/10.4208/cicp.250910.131011a}, Abstract = {The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes. © 2012 Global-Science Press.}, Doi = {10.4208/cicp.250910.131011a}, Key = {fds246894} } @article{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}, Month = {January}, ISSN = {0036-1410}, url = {http://dx.doi.org/10.1137/110839102}, Abstract = {This paper deals with a degenerate diffusion Patlak-Keller-Segel system in n = 3 dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent m = 2n/(n + 2), which is smaller than the usual exponent m* = 2-2/n used in other studies. With the exponent m = 2n/(n + 2), the associated free energy is conformal invariant, and there is a family of stationary solutions Uλ,x0 (x) = C(λ/ λ 2+|x-x0| 2 ) n+2/2 λ < 0, σ0 ? ℝn. For radially symmetric solutions, we prove that if the initial data are strictly below Uλ,0(x) for some λ, then the solution vanishes in L1 loc as tλ8; if the initial data are strictly above Uλ,0(x) for some λ, then the solution either blows up at a finite time or has a mass concentration at r = 0 as time goes to infinity. For general initial data, we prove that there is a global weak solution provided that the Lm norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow-up of the solution if the Lm norm for initial data is larger than the Lm norm of Uλ,x0 (x), which is constant independent of λ and x0, and the free energy of initial data is smaller than that of Uλ,x0(x). © 2012 Society for Industrial and Applied Mathematics.}, Doi = {10.1137/110839102}, Key = {fds246889} } @article{fds220112, Author = {A. Chertock and J.-G. Liu and T. Pendleton}, Title = {Convergence analysis of the particle method for the Camassa-Holm equation}, Pages = {365-373}, Booktitle = {Proceedings of the 13th International Conference on ``Hyperbolic Problems: Theory, Numerics and Applications"}, Publisher = {Higher Education Press}, Address = {Beijing}, Year = {2012}, Key = {fds220112} } @article{fds246887, Author = {Chae, D and Liu, J-G}, Title = {Blow-up, Zero α Limit and the Liouville Type Theorem for the Euler-Poincaré Equations}, Journal = {Communications in Mathematical Physics}, Volume = {314}, Number = {3}, Pages = {671-687}, Year = {2012}, ISSN = {0010-3616}, url = {http://dx.doi.org/10.1007/s00220-012-1534-8}, Abstract = {In this paper we study the Euler-Poincaré equations in ℝ N. We prove local existence of weak solutions in W 2,p(ℝ N),p>N, and local existence of unique classical solutions in H k(ℝ N),k> N/2+3, as well as a blow-up criterion. For the zero dispersion equation (α = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as α → 0, provided that the limiting solution belongs to C([0,T); H k(ℝ N)) with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution u ∈ H 1(ℝ N) is u=0; for α= 0 any weak solution u ∈ L 2(ℝ N) is u=0. © 2012 Springer-Verlag.}, Doi = {10.1007/s00220-012-1534-8}, Key = {fds246887} } @article{fds246888, Author = {Chen, X and Liu, J-G}, Title = {Two nonlinear compactness theorems in L^{p}(0,T;B)}, Journal = {Applied Mathematics Letters}, Volume = {25}, Number = {12}, Pages = {2252-2257}, Year = {2012}, ISSN = {0893-9659}, url = {http://dx.doi.org/10.1016/j.aml.2012.06.012}, Abstract = {We establish two nonlinear compactness theorems in L p(0,T;B) with hypothesis on time translations, which are nonlinear counterparts of two results by Simon (1987) [1]. The first theorem sharpens a result by Maitre (2003) [10] and is important in the study of doubly nonlinear ellipticparabolic equations. Based on this theorem, we then obtain a time translation counterpart of a result by Dubinskiǐ (1965) [5], which is supposed to be useful in the study of some nonlinear kinetic equations (e.g. the FENE-type beadspring chains model). © 2012 Elsevier Ltd. All rights reserved.}, Doi = {10.1016/j.aml.2012.06.012}, Key = {fds246888} } @article{fds246890, Author = {Frouvelle, A and Liu, J-G}, Title = {Dynamics in a kinetic model of oriented particles with phase transition}, Journal = {Siam Journal on Mathematical Analysis}, Volume = {44}, Number = {2}, Pages = {791-826}, Year = {2012}, ISSN = {0036-1410}, url = {http://dx.doi.org/10.1137/110823912}, Abstract = {Motivated by a phenomenon of phase transition in a model of alignment of selfpropelled particles, we obtain a kinetic mean-field equation which is nothing more than the Smoluchowski equation on the sphere with dipolar potential. In this self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global wellposedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, the other equilibria are the Fisher-von Mises distributions with arbitrary direction and a concentration parameter determined by the intensity of the noise. For any initial condition, we give a rigorous proof of convergence of the solution to a steady state as time goes to infinity. In particular, when the noise is under the threshold and with nonzero initial mean velocity, the solution converges exponentially fast to a unique Fisher- von Mises distribution. We also found a new conservation relation, which can be viewed as a convex quadratic entropy when the noise is above the threshold. This provides a uniform exponential rate of convergence to the uniform distribution. At the threshold, we show algebraic decay to the uniform distribution. © 2012 Society for Industrial and Applied Mathematics.}, Doi = {10.1137/110823912}, Key = {fds246890} } @article{fds246891, Author = {Carrillo, JA and Chen, L and Liu, J-G and Wang, J}, Title = {A note on the subcritical two dimensional Keller-Segel system}, Journal = {Acta Applicandae Mathematicae}, Volume = {119}, Number = {1}, Pages = {43-55}, Year = {2012}, ISSN = {0167-8019}, url = {http://dx.doi.org/10.1007/s10440-011-9660-4}, Abstract = {The existence of solution for the 2D-Keller-Segel system in the subcritical case, i.e. when the initial mass is less than 8π, is reproved. Instead of using the entropy in the free energy and free energy dissipation, which was used in the proofs (Blanchet et al. in SIAM J. Numer. Anal. 46:691-721, 2008; Electron. J. Differ. Equ. Conf. 44:32, 2006 (electronic)), the potential energy term is fully utilized by adapting Delort's theory on 2D incompressible Euler equation (Delort in J. Am. Math. Soc. 4:553-386, 1991). © 2011 Springer Science+Business Media B.V.}, Doi = {10.1007/s10440-011-9660-4}, Key = {fds246891} } @article{fds246892, Author = {Degond, P and Liu, J-G}, Title = {Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation}, Journal = {Mathematical Models & Methods in Applied Sciences}, Volume = {22}, Number = {SUPPL.1}, Pages = {1114001-18}, Year = {2012}, ISSN = {0218-2025}, url = {http://dx.doi.org/10.1142/S021820251140001X}, Abstract = {We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first-order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly nonlocal interaction, we derive diffusive corrections to the first-order system which lead to the combination of a heat flow of the harmonic map and LandauLifschitzGilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological LandauLifschitzGilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics. © 2012 World Scientific Publishing Company.}, Doi = {10.1142/S021820251140001X}, Key = {fds246892} } @article{fds246893, Author = {Chertock, A and Liu, J-G and Pendleton, T}, Title = {Convergence of a particle method and global weak solutions of a family of evolutionary PDEs}, Journal = {Siam Journal on Numerical Analysis}, Volume = {50}, Number = {1}, Pages = {1-21}, Year = {2012}, ISSN = {0036-1429}, url = {http://dx.doi.org/10.1137/110831386}, Abstract = {The purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established. © 2012 Society for Industrial and Applied Mathematics.}, Doi = {10.1137/110831386}, Key = {fds246893} } @article{fds246895, Author = {Degond, P and Frouvelle, A and Liu, J-G}, Title = {Macroscopic Limits and Phase Transition in a System of Self-propelled Particles}, Journal = {Journal of Nonlinear Science}, Volume = {23}, Number = {3}, Pages = {1-30}, Year = {2012}, ISSN = {0938-8974}, url = {http://dx.doi.org/10.1007/s00332-012-9157-y}, Abstract = {We investigate systems of self-propelled particles with alignment interaction. Compared to previous work (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193-1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012), the force acting on the particles is not normalized, and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space-inhomogeneous extension of (Frouvelle and Liu, Dynamics in a kinetic model of oriented particles with phase transition, 2012), in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a nonlinear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193-1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012). However, the modified normalization of the force gives rise to different convection speeds, and the resulting model may lose its hyperbolicity in some regions of the state space. © 2012 Springer Science+Business Media New York.}, Doi = {10.1007/s00332-012-9157-y}, Key = {fds246895} } @article{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}, Month = {November}, 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{fds246897, Author = {Liu, J-G and Lorz, A}, Title = {A coupled chemotaxis-fluid model: Global existence}, Journal = {Annales De L'Institut Henri Poincare (C) Non Linear Analysis}, Volume = {28}, Number = {5}, Pages = {643-652}, Year = {2011}, Month = {September}, 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{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{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}, 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}, Month = {May}, 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}, Month = {January}, 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. Â© 2010 International Press.}, Key = {fds246928} } @article{fds304584, Author = {Liu, J-G and Pego, RL}, Title = {Stable discretization of magnetohydrodynamics in bounded domains}, Journal = {Communications in Mathematical Sciences}, Volume = {8}, Number = {1}, Pages = {235-251}, Year = {2010}, ISSN = {1539-6746}, Abstract = {We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with a perfectly conducting boundary. In the scheme, the velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently de-veloped formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray projection operators. We prove stability of the time-difference scheme, and deduce a local-time well-posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so. © 2010 International Press.}, Key = {fds304584} } @article{fds246943, Author = {Liu, J-G and Liu, J and Pego, RL}, Title = {Error estimates for finite-element Navier-Stokes solvers without standard Inf-Sup conditions}, Journal = {Chinese Annals of Mathematics, Series B}, Volume = {30}, Number = {6}, Pages = {743-768}, Year = {2009}, ISSN = {0252-9599}, url = {http://dx.doi.org/10.1007/s11401-009-0116-3}, Abstract = {The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C1 elements for velocity and C0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis. © Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2009.}, Doi = {10.1007/s11401-009-0116-3}, Key = {fds246943} } @article{fds246944, Author = {Liu, J-G and Wang, W-C}, Title = {Characterization and regularity for axisymmetric solenoidal vector fields with application to navier-stokes equation}, Journal = {Siam Journal on Mathematical Analysis}, Volume = {41}, Number = {5}, Pages = {1825-1850}, Year = {2009}, ISSN = {0036-1410}, url = {http://dx.doi.org/10.1137/080739744}, Abstract = {We consider the vorticity-stream formulation of axisymmetric incompressible flows and its equivalence with the primitive formulation. It is shown that, to characterize the regularity of a divergence free axisymmetric vector field in terms of the swirling components, an extra set of pole conditions is necessary to give a full description of the regu larity. In addition, smooth solutions up to the axis of rotation give rise to smooth solutions of primitive formulation in the case of the Navier-Stokes equation, but not the Euler equation. We also establish a proper weak formulation and show its equivalence to Leray's formulation. © 2009 Society for Industrial and Applied Mathematics.}, Doi = {10.1137/080739744}, Key = {fds246944} } @article{fds246945, Author = {Ha, S-Y and Liu, J-G}, Title = {A simple proof of the Cucker-Smale flocking dynamics and mean-field limit}, Journal = {Communications in Mathematical Sciences}, Volume = {7}, Number = {2}, Pages = {297-325}, Year = {2009}, ISSN = {1539-6746}, url = {http://dx.doi.org/10.4310/CMS.2009.v7.n2.a2}, 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.}, Doi = {10.4310/CMS.2009.v7.n2.a2}, Key = {fds246945} } @article{fds246940, Author = {Hsia, C-H and Liu, J-G and Wang, C}, Title = {Structural stability and bifurcation for 2D incompressible ows with symmetry}, Journal = {Meth. Appl. Anal.}, Volume = {15}, Pages = {495-512}, Year = {2008}, Key = {fds246940} } @article{fds246941, Author = {Lin, P and Liu, J-G and Lu, X}, Title = {Long time numerical solution of the Navier-Stokes equations based on a sequential regularization formulation}, Journal = {Siam Journal on Scientific Computing}, Volume = {31}, Number = {1}, Pages = {398-419}, Year = {2008}, ISSN = {1064-8275}, url = {http://dx.doi.org/10.1137/060673722}, Abstract = {The sequential regularization method is a reformulation of the unsteady Navier-Stokes equations from the viewpoint of constrained dynamical systems or the approximate Helmholtz-Hodge projection. In this paper we study the long time behavior of the sequential regularization formulation. We give a uniform-in-time estimate between the solution of the reformulated system and that of the Navier-Stokes equations. We also conduct an error analysis for the temporal discrete system and show that the error bound is independent of time. A couple of long time flow examples are computed to demonstrate this method. © 2008 Society for Industrial and Applied Mathematics.}, Doi = {10.1137/060673722}, Key = {fds246941} } @article{fds246942, Author = {Liu, J-G and Wang, C}, Title = {A fourth order numerical method for the primtive equations formulated in mean vorticity}, Journal = {Communications in Computational Physics}, Volume = {4}, Number = {1}, Pages = {26-55}, Year = {2008}, ISSN = {1815-2406}, Abstract = {A fourth-order finite difference method is proposed and studied for the primitive equations (PEs) of large-scale atmospheric and oceanic flow based on mean vorticity formulation. Since the vertical average of the horizontal velocity field is divergence-free, we can introduce mean vorticity and mean stream function which are connected by a 2-D Poisson equation. As a result, the PEs can be reformulated such that the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of the horizontal velocity. The mean vorticity equation is approximated by a compact difference scheme due to the difficulty of the mean vorticity boundary condition, while fourth-order long-stencil approximations are utilized to deal with transport type equations for computational convenience. The numerical values for the total velocity field (both horizontal and vertical) are statically determined by a discrete realization of a differential equation at each fixed horizontal point. The method is highly efficient and is capable of producing highly resolved solutions at a reasonable computational cost. The full fourth-order accuracy is checked by an example of the reformulated PEs with force terms. Additionally, numerical results of a large-scale oceanic circulation are presented. © 2008 Global-Science Press.}, Key = {fds246942} } @article{fds246946, Author = {Degond, P and Liu, J-G and Vignal, M-H}, Title = {Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit}, Journal = {Siam Journal on Numerical Analysis}, Volume = {46}, Number = {3}, Pages = {1298-1322}, Year = {2008}, ISSN = {0036-1429}, url = {http://dx.doi.org/10.1137/070690584}, Keywords = {stiffness • Debye length • electron plasma period • Burgers-Poisson • sheath problem • Klein-Gordon}, Abstract = {In a previous work [P. Crispel, P. Degond, and M.-H. Vignal, J. Comput. Phys., 223 (2007), pp. 208-234], a new numerical discretization of the Euler-Poisson system was proposed. This scheme is "asymptotic preserving" in the quasineutral limit (i.e., when the Debye length ε tends to zero), which means that it becomes consistent with the limit model when ε → 0. In the present work, we show that the stability domain of the present scheme is independent of ε. This stability analysis is performed on the Fourier transformed (with respect to the space variable) linearized system. We show that the stability property is more robust when a space-decentered scheme is used (which brings in some numerical dissipation) rather than a space-centered scheme. The linearization is first performed about a zero mean velocity and then about a nonzero mean velocity. At the various stages of the analysis, our scheme is compared with more classical schemes and its improved stability property is outlined. The analysis of a fully discrete (in space and time) version of the scheme is also given. Finally, some considerations about a model nonlinear problem, the Burgers-Poisson problem, are also discussed. © 2008 Society for Industrial and Applied Mathematics.}, Doi = {10.1137/070690584}, Key = {fds246946} } @article{fds246948, Author = {Lu, X and Lin, P and Liu, J-G}, Title = {Analysis of a sequential regularization method for the unsteady Navier-Stokes equations}, Journal = {Mathematics of Computation}, Volume = {77}, Number = {263}, Pages = {1467-1494}, Year = {2008}, ISSN = {0025-5718}, url = {http://dx.doi.org/10.1090/S0025-5718-08-02087-5}, Keywords = {Navier-Stokes equations • iterative penalty method • implicit parabolic PDE • error estimates • constrained dynamical system • stabilization method}, Abstract = {The incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods. In the paper, we study the existence and uniqueness for the solution of the SRM and provide a simple proof of the convergence of the solution of the SRM to the solution of the Navier-Stokes equations. We also give error estimates for the time discretized SRM formulation. ©2008 American Mathematical Society.}, Doi = {10.1090/S0025-5718-08-02087-5}, Key = {fds246948} } @article{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 and Applied Mathematics}, Volume = {60}, Number = {10}, Pages = {1443-1487}, Year = {2007}, Month = {October}, ISSN = {0010-3640}, url = {http://dx.doi.org/10.1002/cpa.20178}, Abstract = {For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf-sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier-Stokes equations as a perturbed diffusion equation. ©2006 Wiley Periodicals, Inc.}, Doi = {10.1002/cpa.20178}, Key = {fds246880} } @article{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{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{fds246957, Author = {Moore, J and Liu, J-G and Zhou, K and Yu, LL}, 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}, Month = {July}, 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 (GxE) 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.}, Doi = {10.1021/jf060381l}, Key = {fds246957} } @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{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{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}, Month = {October}, ISSN = {0938-8974}, url = {http://dx.doi.org/10.1007/s00332-004-0634-9}, Abstract = {We study a continuum model for epitaxial growth of thin films in which the slope of mound structure of film surface increases. This model is a diffusion equation for the surface height profile h which is assumed to satisfy the periodic boundary condition. The equation happens to possess a Liapunov or "free-energy" functional. This functional consists of the term |Δ h| 2, which represents the surface diffusion, and-log (1 + |∇ h| 2), which describes the effect of kinetic asymmetry in the adatom attachment-detachment. We first prove for large time t that the interface width-the standard deviation of the height profile-is bounded above by O(t 1/2), the averaged gradient is bounded above by O(t 1/4), and the averaged energy is bounded below by O(-log t). We then consider a small coefficient ε 2 of |Δ h| 2 with ε = 1/L and L the linear size of the underlying system, and study the energy asymptotics in the large system limit ε → 0. We show that global minimizers of the free-energy functional exist for each ε > 0, the L 2-norm of the gradient of any global minimizer scales as O(1/ε), and the global minimum energy scales as O( log ε). The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelengths. Finally, we apply our minimum energy estimates to derive bounds in terms of the linear system size L for the saturation interface width and the corresponding saturation time. © 2005 Springer.}, Doi = {10.1007/s00332-004-0634-9}, Key = {fds304585} } @article{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}, Month = {September}, 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{fds246902, Author = {Liu, J-G and Wang, W-C}, Title = {Energy and helicity preserving schemes for hydro- and magnetohydro-dynamics flows with symmetry}, Journal = {Journal of Computational Physics}, Volume = {200}, Number = {1}, Pages = {8-33}, Year = {2004}, url = {http://dx.doi.org/10.1016/j.jcp.2004.03.005}, Abstract = {We propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry. By introducing a generalized vorticity-stream formulation, the divergence free constraints are automatically satisfied. In addition, with explicit treatment of the nonlinear terms and local vorticity boundary condition, the Navier-Stokes (MHD, respectively) equation essentially decouples into 2 (4, respectively) scalar equation and thus the scheme is very efficient. Moreover, with proper discretization of the nonlinear terms, the scheme preserves both energy and helicity identities numerically. This is achieved by recasting the nonlinear terms (convection, vorticity stretching, geometric source, Lorentz force and electro-motive force) in terms of Jacobians. This conservative property is valid even in the presence of the pole singularity for axisymmetric flows. The exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity. Numerical examples have demonstrated both accuracy and efficiency of the scheme. Finally, local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost. © 2004 Elsevier Inc. All rights reserved.}, Doi = {10.1016/j.jcp.2004.03.005}, Key = {fds246902} } @article{fds246954, Author = {Lin, H-E and Liu, J-G and Xu, W-Q}, Title = {Effects of small viscosity and far field boundary conditions for hyperbolic systems}, Journal = {Communications on Pure and Applied Analysis}, Volume = {3}, Number = {2}, Pages = {267-290}, Year = {2004}, ISSN = {1534-0392}, Abstract = {In this paper we study the effects of small viscosity term and the far-field boundary conditions for systems of convection-diffusion equations in the zero viscosity limit. The far-field boundary conditions are classified and the corresponding solution structures are analyzed. It is confirmed that the Neumann type of far-field boundary condition is preferred. On the other hand, we also identify a class of improperly coupled boundary conditions which lead to catastrophic reflection waves dominating the inlet in the zero viscosity limit. The analysis is performed on the linearized convection-diffusion model which well describes the behavior at the far field for many physical and engineering systems such as fluid dynamical equations and electro-magnetic equations. The results obtained here should provide some theoretical guidance for designing effective far field boundary conditions.}, Key = {fds246954} } @article{fds246955, Author = {Liu, J-G and Xu, W-Q}, Title = {Far field boundary condition for convection diffusion equation at zero viscosity limit}, Journal = {Quarterly of Applied Mathematics}, Volume = {62}, Number = {1}, Pages = {27-52}, Year = {2004}, Abstract = {In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by the viscosity limit as a solution of the inviscid equation. The results confirm that the Neumann type of far-field boundary condition is preferred in the outlet and characteristic boundary dondition. Under some appropriate regularity and compatibility conditions on the initial and boundary data, we obtain optimal error estimates between the full viscous solution and the inviscid solution with suitable boundary layer corrections. These results hold in arbitrary space dimensions and similar statements also hold for the strip problem This model well describes the behavior at the far-field for many physical and engineering systems such as fluid dynamical equation and electro-magnetic equation. The results obtained here should provide some theoretical guidance for designing effective far-field boundary conditions.}, Key = {fds246955} } @article{fds246956, Author = {Wang, C and Liu, J-G and Johnston, H}, Title = {Analysis of a fourth order finite difference method for the incompressible Boussinesq equations}, Journal = {Numerische Mathematik}, Volume = {97}, Number = {3}, Pages = {555-594}, Year = {2004}, url = {http://dx.doi.org/10.1007/s00211-003-0508-3}, Abstract = {The convergence of a fourth order finite difference method for the 2-D unsteady, viscous incompressible Boussinesq equations, based on the vorticity-stream function formulation, is established in this article. A compact fourth order scheme is used to discretize the momentum equation, and long-stencil fourth order operators are applied to discretize the temperature transport equation. A local vorticity boundary condition is used to enforce the no-slip boundary condition for the velocity. One-sided extrapolation is used near the boundary, dependent on the type of boundary condition for the temperature, to prescribe the temperature at "ghost" points lying outside of the computational domain. Theoretical results of the stability and accuracy of the method are also provided. In numerical experiments the method has been shown to be capable of producing highly resolved solutions at a reasonable computational cost.}, Doi = {10.1007/s00211-003-0508-3}, Key = {fds246956} } @article{fds246959, Author = {Li, B and Liu, J-G}, Title = {Eptaxial growth without slope selection: energetics, coarsening, and dynamic scaling}, Journal = {J. Nonlinear Sci.}, Volume = {14}, Number = {5}, Pages = {429-451}, Year = {2004}, ISSN = {0938-8974}, url = {http://dx.doi.org/10.1007/s00332-004-0634-9}, Abstract = {We study a continuum model for epitaxial growth of thin films in which the slope of mound structure of film surface increases. This model is a diffusion equation for the surface height profile h which is assumed to satisfy the periodic boundary condition. The equation happens to possess a Liapunov or "free-energy" functional. This functional consists of the term |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 > 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 saturation time. Â© 2005 Springer.}, Doi = {10.1007/s00332-004-0634-9}, Key = {fds246959} } @article{fds246963, Author = {Ghil, M and Liu, J-G and Wang, C and Wang, S}, Title = {Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow}, Journal = {Physica D: Nonlinear Phenomena}, Volume = {197}, Number = {1-2}, Pages = {149-173}, Year = {2004}, ISSN = {0167-2789}, url = {http://dx.doi.org/10.1016/j.physd.2004.06.012}, Abstract = {We study the detailed process of bifurcation in the flow's topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticily formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present. © 2004 Elsevier B.V. All rights reserved.}, Doi = {10.1016/j.physd.2004.06.012}, Key = {fds246963} } @article{fds246965, Author = {Liu, J-G and Wang, C}, Title = {High order finite difference method for unsteady incompressible flow on multi-connected domain in vorticity-stream function formulation}, Journal = {Computer and Fluids}, Volume = {33}, Number = {2}, Pages = {223-255}, Year = {2004}, url = {http://dx.doi.org/10.1016/S0045-7930(03)00037-9}, Abstract = {Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order schemes can be efficiently implemented. However, a difficulty arises in a multi-connected computational domain in determining the constants for the stream function on the boundary of the "holes". This is an especially challenging task for the calculation of unsteady flows, since these constants vary with time to reflect the total fluxes of the flow in each sub-channel. In this paper, we propose an efficient method in a finite difference setting to solve this problem and present some numerical experiments, including an accuracy check of a Taylor vortex-type flow, flow past a non-symmetric square, and flow in a heat exchanger. Â© 2003 Elsevier Ltd. All rights reserved.}, Doi = {10.1016/S0045-7930(03)00037-9}, Key = {fds246965} } @article{fds304583, Author = {Liu, J-G and Wang, C}, Title = {High order finite difference methods for unsteady incompressible flows in multi-connected domains}, Journal = {Computers and Fluids}, Volume = {33}, Number = {2}, Pages = {223-255}, Year = {2004}, url = {http://dx.doi.org/10.1016/S0045-7930(03)00037-9}, Abstract = {Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order schemes can be efficiently implemented. However, a difficulty arises in a multi-connected computational domain in determining the constants for the stream function on the boundary of the "holes". This is an especially challenging task for the calculation of unsteady flows, since these constants vary with time to reflect the total fluxes of the flow in each sub-channel. In this paper, we propose an efficient method in a finite difference setting to solve this problem and present some numerical experiments, including an accuracy check of a Taylor vortex-type flow, flow past a non-symmetric square, and flow in a heat exchanger. © 2003 Elsevier Ltd. All rights reserved.}, Doi = {10.1016/S0045-7930(03)00037-9}, Key = {fds304583} } @article{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}, Month = {December}, 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{fds246968, Author = {Liu, JG 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}, Month = {April}, 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{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{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{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}, Month = {July}, 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{fds246937, Author = {Wang, C and Liu, J-G}, Title = {Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation}, Journal = {Numerische Mathematik}, Volume = {91}, Number = {3}, Pages = {543-576}, Year = {2002}, url = {http://dx.doi.org/10.1007/s002110100311}, Abstract = {In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thorn's formula, Wilkes' formula, or other local formulas in the earlier literature can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function. We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth order scheme using a 1-D Stokes model.}, Doi = {10.1007/s002110100311}, Key = {fds246937} } @article{fds246938, Author = {Weinan, E and Liu, J-G}, Title = {Projection method III: Spatial discretization on the staggered grid}, Journal = {Mathematics of Computation}, Volume = {71}, Number = {237}, Pages = {27-47}, Year = {2002}, url = {http://dx.doi.org/10.1090/S0025-5718-01-01313-8}, Abstract = {In E & Liu (SIAM J Numer. Anal., 1995), we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous (we call this the semi-discrete case). In this paper, we address similar questions for the fully discrete case when the spatial variables are discretized using a staggered grid. We prove that the numerical solution in velocity has full accuracy up to the boundary, despite the fact that there are numerical boundary layers present in the semi-discrete solutions.}, Doi = {10.1090/S0025-5718-01-01313-8}, Key = {fds246938} } @article{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{fds246933, Author = {Weinan, E and Liu, JG}, 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}, Month = {December}, ISSN = {0271-2091}, url = {http://dx.doi.org/10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B}, Abstract = {A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E. Liu J-G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher-order) finite elements. This method can achieve high-order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite elements methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.}, Doi = {10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B}, Key = {fds246933} } @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}, Month = {May}, url = {http://dx.doi.org/10.1006/jcph.2000.6475}, Abstract = {In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments. © 2000 Academic Press.}, Doi = {10.1006/jcph.2000.6475}, Key = {fds246931} } @article{fds246930, Author = {Liu, JG 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}, Month = {January}, url = {http://dx.doi.org/10.1002/(SICI)1097-0312(200006)53:6<786::AID-CPA3>3.0.CO;2-Y}, Abstract = {We prove the convergence of a discontinuous Galerkin method approximating the 2-D incompressible Euler equations with discontinuous initial vorticity: ω0∈ L2(Ω). Furthermore, when ω0∈ L∞(Ω), the whole sequence is shown to be strongly convergent. This is the first convergence result in numerical approximations of this general class of discontinuous flows. Some important flows such as vortex patches belong to this class. © 2000 John Wiley & Sons, Inc.}, Doi = {10.1002/(SICI)1097-0312(200006)53:6<786::AID-CPA3>3.0.CO;2-Y}, Key = {fds246930} } @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{fds246927, Author = {Lefloch, PG and Liu, J-G}, Title = {Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions}, Journal = {Mathematics of Computation}, Volume = {68}, Number = {227}, Pages = {1025-1055}, Year = {1999}, Abstract = {Solutions of conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this property from a numerical standpoint. We introduce a class of fully discrete in space and time, high order accurate, difference schemes, called generalized monotone schemes. Convergence toward the entropy solution is proven via a new technique of proof, assuming that the initial data has a finite number of extremum values only, and the flux-function is strictly convex. We define discrete paths of extrema by tracking local extremum values in the approximate solution. In the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum. As a corollary, we obtain a proof of convergence for a MUSCL-type scheme that is second order accurate away from sonic points and extrema.}, Key = {fds246927} } @article{fds246929, Author = {Wang, ZJ and Liu, JG and Childress, S}, Title = {Connection between corner vortices and shear layer instability in flow past an ellipse}, Journal = {Physics of Fluids}, Volume = {11}, Number = {9}, Pages = {2446-2448}, Year = {1999}, Abstract = {We investigate, by numerical simulation, the shear layer instability associated with the outer layer of a spiral vortex formed behind an impulsively started thin ellipse. The unstable free shear layer undergoes a secondary instability. We connect this instability with the dynamics of corner vortices adjacent to the tip of the ellipse by observing that the typical turnover time of the corner vortex matches the period of the unstable mode in the shear layer. We suggest that the corner vortex acts as a signal generator, and produces periodic perturbation which triggers the instability. © 1999 American Institute of Physics.}, Key = {fds246929} } @article{fds246925, Author = {Xu, E and Liu, J-G}, Title = {Pricing of mortgage-backed securities with option-adjusted spread}, Journal = {Managerial Finance}, Volume = {24}, Pages = {94-109}, Year = {1998}, Key = {fds246925} } @article{fds246926, Author = {Choi, H and Liu, J-G}, Title = {The Reconstruction of Upwind Fluxes for Conservation Laws: Its Behavior in Dynamic and Steady State Calculations}, Journal = {Journal of Computational Physics}, Volume = {144}, Number = {2}, Pages = {237-256}, Year = {1998}, url = {http://dx.doi.org/10.1006/jcph.1998.5970}, Abstract = {The Euler equation of compressible flows is solved by the finite volume method, where high order accuracy is achieved by the reconstruction of each component of upwind fluxes of a flux splitting using the biased averaging procedure. Compared to the solution reconstruction in Godunov-type methods, its implementation is simple and easy, and the computational complexity is relatively low. This approach is parameter-free and requires neither a Riemann solver nor field-by-field decomposition. The numerical results from both dynamic and steady state calculations demonstrate the accuracy and robustness of this approach. Some techniques for the acceleration of the convergence to the steady state are discussed, including multigrid and multistage Runge-Kutta time methods. © 1998 Academic Press.}, Doi = {10.1006/jcph.1998.5970}, Key = {fds246926} } @article{fds246922, Author = {Weinan, E and Liu, J-G}, Title = {Finite Difference Methods for 3D Viscous Incompressible Flows in the Vorticity-Vector Potential Formulation on Nonstaggered Grids}, Journal = {Journal of Computational Physics}, Volume = {138}, Number = {1}, Pages = {57-82}, Year = {1997}, url = {http://dx.doi.org/10.1006/jcph.1997.5815}, Abstract = {Simple, efficient, and accurate finite difference methods are introduced for 3D unsteady viscous incompressible flows in the vorticity-vector potential formulation on nonstaggered grids. Two different types of methods are discussed. They differ in the implementation of the normal component of the vorticity boundary condition and consequently the enforcement of the divergence free condition for vorticity. Both second-order and fourth-order accurate schemes are developed. A detailed accuracy test is performed, revealing the structure of the error and the effect of how the convective terms are discretized near the boundary. The influence of the divergence free condition for vorticity to the overall accuracy is studied. Results on the cubic driven cavity flow at Reynolds number 500 and 3200 are shown and compared with that of the MAC scheme. © 1997 Academic Press.}, Doi = {10.1006/jcph.1997.5815}, Key = {fds246922} } @article{fds246923, Author = {Chen, G-Q and Liu, J-G}, Title = {Convergence of difference schemes with high resolution for conservation laws}, Journal = {Mathematics of Computation}, Volume = {66}, Number = {219}, Pages = {1027-1053}, Year = {1997}, Abstract = {We are concerned with the convergence of Lax-Weridroff type schemes with high resolution to the entropy solutions fo: conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions-the Richtrayer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth flux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed Lp estimates of the approximate solutions, H-1 compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws.}, Key = {fds246923} } @article{fds246924, Author = {Weinan, E and Liu, J-G}, Title = {Finite difference schemes for incompressible flows in the velocity - impulse density formulation}, Journal = {Journal of Computational Physics}, Volume = {130}, Number = {1}, Pages = {67-76}, Year = {1997}, Abstract = {We consider finite difference schemes based on the impulse density variable. We show that the original velocity - impulse density formulation of Oseledets is marginally ill-posed for the inviscid flow, and this has the consequence that some ordinarily stable numerical methods in other formulations become unstable in the velocity - impulse density formulation. We present numerical evidence of this instability. We then discuss the construction of stable finite difference schemes by requiring that at the numerical level the nonlinear terms be convertible to similar terms in the primitive variable formulation. Finally we give a simplified velocity - impulse density formulation which is free of these complications and yet retains the nice features of the original velocity - impulse density formulation with regard to the treatment of boundary. We present numerical results on this simplified formulation for the driven cavity flow on both the staggered and non-staggered grids. © 1997 Academic Press.}, Key = {fds246924} } @article{fds246914, Author = {Jin, S and Liu, J-G}, Title = {Oscillations induced by numerical viscosities}, Journal = {Mat. Contemp.}, Volume = {10}, Pages = {169-180}, Year = {1996}, Key = {fds246914} } @article{fds246915, Author = {Jin, S and Liu, J-G}, Title = {The effects of numerical viscosities: I. Slowly moving shocks}, Journal = {Journal of Computational Physics}, Volume = {126}, Number = {2}, Pages = {373-389}, Year = {1996}, url = {http://dx.doi.org/10.1006/jcph.1996.0144}, Abstract = {We begin a systematical study on the effect of numerical viscosities. In this paper we investigate the behavior of shock-capturing methods for slowly moving shocks. It is known that for slowly moving shocks even a first-order scheme, such as the Godunov or Roe type methods, will generate downstream oscillatory wave patterns that cannot be effectively damped by the dissipation of these first-order schemes. The purpose of this paper is to understand the formation and behavior of these downstream patterns. Our study shows that the downstream errors are generated by the unsteady nature of the viscous shock profiles and behave diffusively. The scenario is as follows. When solving the compressible Euler equations by shock capturing methods, the smeared density profile introduces a momentum spike at the shock location if the shock moves slowly. Downstream waves will necessarily emerge in order to balance the momentum mass carried by the spike for the momentum conservation. Although each family of waves decays in l∞ and l2 while they preserve the same mass, the perturbing nature of the viscous or spike profile is a constant source for the generation of new downstream waves, causing spurious solutions for all time. Higher order TVD or ENO type interpolations accentuate this problem. © 1996 Academic Press, Inc.}, Doi = {10.1006/jcph.1996.0144}, Key = {fds246915} } @article{fds246916, Author = {Weinan, E and Liu, J-G}, Title = {Vorticity boundary condition and related issues for finite difference schemes}, Journal = {Journal of Computational Physics}, Volume = {124}, Number = {2}, Pages = {368-382}, Year = {1996}, url = {http://dx.doi.org/10.1006/jcph.1996.0066}, Abstract = {This paper discusses three basic issues related to the design of finite difference schemes for unsteady viscous incompressible flows using vorticity formulations: the boundary condition for vorticity, an efficient time-stepping procedure, and the relation between these schemes and the ones based on velocity-pressure formulation. We show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier. We also show that if we couple a standard centered difference scheme with third-or fourth-order explicit Runge-Kutta methods, the resulting schemes have no cell Reynolds number constraints. For high Reynolds number flows, these schemes are stable under the CFL condition given by the convective terms. Finally, we show that the classical MAC scheme is the same as Thom's formula coupled with second-order centered differences in the interior, in the sense that one can define discrete vorticity in a natural way for the MAC scheme and get the same values as the ones computed from Thom's formula. We use this to derive an efficient fourth-order Runge-Kutta time discretization for the MAC scheme from the one for Thom's formula. We present numerical results for driven cavity flow at high Reynolds number (105). © 1996 Academic Press, Inc.}, Doi = {10.1006/jcph.1996.0066}, Key = {fds246916} } @article{fds246917, Author = {Weinan, E and Liu, J-G}, Title = {Essentially compact schemes for unsteady viscous incompressible flows}, Journal = {Journal of Computational Physics}, Volume = {126}, Number = {1}, Pages = {122-138}, Year = {1996}, url = {http://dx.doi.org/10.1006/jcph.1996.0125}, Abstract = {A new fourth-order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced. The scheme is based on the vorticity-stream function formulation. It is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions. It is also very efficient, at every time step or Runge-Kutta stage, only two Poisson-like equations have to be solved. The Poisson-like equations are amenable to standard fast Poisson solvers usually designed for second order schemes. Detailed comparison with the second-order scheme shows the clear superiority of this new fourth-order scheme in resolving both the boundary layers and the gross features of the flow. This efficient fourth-order scheme also made it possible to compute the driven cavity flow at Reynolds number 106 on a 10242 grid at a reasonable cost. Fourth-order convergence is proved under mild regularity requirements. This is the first such result to our knowledge. © 1996 Academic Press, Inc.}, Doi = {10.1006/jcph.1996.0125}, Key = {fds246917} } @article{fds246918, Author = {Weinan, E and Liu, J-G}, Title = {Projection method II: Godunov-Ryabenki analysis}, Journal = {SIAM Journal on Numerical Analysis}, Volume = {33}, Number = {4}, Pages = {1597-1621}, Year = {1996}, Abstract = {This is the second of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of the present paper is to explain why the accuracy of the velocity approximation is not affected by (1) the numerical boundary layers in the approximation of pressure and the intermediate velocity field and (2) the noncommutativity of the projection operator and the laplacian. This is done by using a Godunov-Ryabenki type of analysis in a rigorous fashion. By doing so, we hope to be able to convey the message that normal mode analysis is basically sufficient for understanding the stability and accuracy of a finite-difference method for the Navier-Stokes equation even in the presence of boundaries. As an example, we analyze the second-order projection method based on pressure increment formulations used by van Kan and Bell, Colella, and Glaz. The leading order error term in this case is of O(Δt) and behaves as high frequency oscillations over the whole domain, compared with the O(Δt1/2) numerical boundary layers found in the second-order Kim-Moin method.}, Key = {fds246918} } @article{fds246919, Author = {Levermore, CD and Liu, J-G}, Title = {Large oscillations arising in a dispersive numerical scheme}, Journal = {Physica D: Nonlinear Phenomena}, Volume = {99}, Number = {2-3}, Pages = {191-216}, Year = {1996}, Abstract = {We study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the Hopf equation whenever the classical solution of that equation develops a singularity. Modulation equations are derived that describe period-two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic. However, those equations have an elliptic region that may be entered by its solutions in a unite time, after which the corresponding period-two oscillations are seen to break down. This kind of phenomenon has not been observed for integrable schemes. The generation and propagation of period-two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions. Modulation equations are also presented for period-three oscillations. Numerical experiments are carried out that illustrate our analysis. © 1996 Elsevier Science B.V. All rights reserved.}, Key = {fds246919} } @article{fds246920, Author = {Liu, J-G and Xin, Z}, Title = {Kinetic and viscous boundary layers for broadwell equations}, Journal = {Transport Theory and Statistical Physics}, Volume = {25}, Number = {3-5}, Pages = {447-461}, Year = {1996}, Abstract = {In this paper, we investigate the boundary layer behavior of solutions to the one dimensional Broadwell model of the nonlinear Boltzmann equation for small mean free path. We consider the analogue of Maxwell's diffusive and the reflexive boundary conditions. It is found that even for such a simple model, there are boundary layers due to purely kinetic effects which cannot be detected by the corresponding Navier-Stokes system. It is also found numerically that a compressive boundary layer is not always stable in the sense that it may detach from the boundary and move into the interior of the gas as a shock layer.}, Key = {fds246920} } @article{fds246921, Author = {Liu, J-G and Xin, Z}, Title = {Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation}, Journal = {Arch. Rat. Mech. Anal.}, Volume = {135}, Pages = {61-105}, Year = {1996}, Key = {fds246921} } @article{fds246912, Author = {Weinan, E and Liu, J-G}, Title = {Projection method I: convergence and numerical boundary layers}, Journal = {SIAM J. Numer. Anal.}, Volume = {32}, Pages = {1017-1057}, Year = {1995}, Key = {fds246912} } @article{fds246913, Author = {Liu, J-G and Xin, Z}, Title = {Convergence of vortex methods for weak solutions to the 2-D Euler equations with vortex sheets data}, Journal = {Comm. Pure Appl. Math.}, Volume = {48}, Pages = {611-628}, Year = {1995}, Key = {fds246913} } @article{fds246910, Author = {Lefloch, P and Liu, J-G}, Title = {Discrete entropy and monotonicity criteria for hyperbolic conservation laws}, Journal = {C.R. Acad. Sci. Paris.}, Volume = {319}, Pages = {881-886}, Year = {1994}, Key = {fds246910} } @article{fds246911, Author = {Jin, S and Liu, J-G}, Title = {Relaxation and diffusion enhanced dispersive waves}, Journal = {Proceedings of The Royal Society of London, Series A: Mathematical and Physical Sciences}, Volume = {446}, Number = {1928}, Pages = {555-563}, Year = {1994}, Abstract = {The development of shocks in nonlinear hyperbolic conservation laws may be regularized through either diffusion or relaxation. However, we have observed surprisingly that for some physical problems, when both of the smoothing factors diffusion and relaxation coexist, under appropriate asymptotic assumptions, the dispersive waves are enhanced. This phenomenon is studied asymptotically in the sense of the Chapman-Enskog expansion and demonstrated numerically.}, Key = {fds246911} } @article{fds246906, Author = {Chen, G-Q and Liu, J-G}, Title = {Convergence of second-order schemes for isentropic gas dynamics}, Journal = {Math. Comp.}, Volume = {61}, Pages = {607-629}, Year = {1993}, Key = {fds246906} } @article{fds246907, Author = {Engquist, B and Liu, J-G}, Title = {Numerical methods for oscillatory solutions to hyperbolic problems}, Journal = {Comm. Pure Appl. Math.}, Volume = {46}, Pages = {1327-1361}, Year = {1993}, Key = {fds246907} } @article{fds246908, Author = {Liu, J-G and Xin, Z}, Title = {L1-stability of stationary discrete shocks}, Journal = {Math. Comp.}, Volume = {60}, Pages = {233-244}, Year = {1993}, Key = {fds246908} } @article{fds246909, Author = {Liu, J-G and Xin, Z}, Title = {Nonlinear stability of discrete shocks for systems of conservation laws}, Journal = {Archive for Rational Mechanics and Analysis}, Volume = {125}, Number = {3}, Pages = {217-256}, Year = {1993}, ISSN = {0003-9527}, url = {http://dx.doi.org/10.1007/BF00383220}, Abstract = {In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m×m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the Lp-norm for all p ≧ 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in Lp (P ≧ 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme. © 1993 Springer-Verlag.}, Doi = {10.1007/BF00383220}, Key = {fds246909} } %% Papers Accepted @article{fds325700, Author = {Degond, P and Liu, J-G and Pego, RL}, Title = {Coagulation–Fragmentation Model for Animal Group-Size Statistics}, Journal = {Journal of Nonlinear Science}, Volume = {27}, Number = {2}, Pages = {379-424}, Year = {2017}, Month = {April}, url = {http://dx.doi.org/10.1007/s00332-016-9336-3}, Doi = {10.1007/s00332-016-9336-3}, Key = {fds325700} } @article{fds327636, Author = {Huang, H and Liu, J-G}, Title = {Error estimate of a random particle blob method for the Keller-Segel equation}, Journal = {Mathematics of Computation}, Volume = {86}, Number = {308}, Pages = {2719-2744}, Year = {2017}, Month = {February}, url = {http://dx.doi.org/10.1090/mcom/3174}, Doi = {10.1090/mcom/3174}, Key = {fds327636} } @article{fds325701, Author = {Liu, J-G and Wang, J}, Title = {Global existence for a thin film equation with subcritical mass}, Journal = {Discrete and Continuous Dynamical Systems Series B}, Volume = {22}, Number = {4}, Pages = {1461-1492}, Year = {2017}, Month = {February}, url = {http://dx.doi.org/10.3934/dcdsb.2017070}, Doi = {10.3934/dcdsb.2017070}, Key = {fds325701} } @article{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} }