Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method

We focus on mono-dimensional hyperbolic systems approximated by a particular lattice Boltzmann scheme. The scheme is described in the framework of the multiple relaxation times method and stability conditions are given. An analysis is done to link the scheme with an explicit finite differences approximation of the relaxation method proposed by Jin and Xin. Several numerical illustrations are given for the transport equation, Burger's equation, the p-system, and full compressible Euler's system.

[1]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[2]  Paul J. Dellar,et al.  Lattice and discrete Boltzmann equations for fully compressible flow , 2006 .

[3]  V. Giovangigli.  Multicomponent flow modeling , 1999 .

[4]  Michael Junk,et al.  A finite difference interpretation of the lattice Boltzmann method , 2001 .

[5]  Pierre Lallemand,et al.  Lattice Boltzmann simulations of thermal convective flows in two dimensions , 2013, Comput. Math. Appl..

[6]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[7]  Centro internazionale matematico estivo. Session,et al.  Advanced Numerical Approximation of Nonlinear Hyperbolic Equations , 1998 .

[8]  F. Lagoutière,et al.  Probabilistic Analysis of the Upwind Scheme for Transport Equations , 2007, 0712.3217.

[9]  Xijun Yu,et al.  Lattice Boltzmann modeling and simulation of compressible flows , 2012, 1209.3542.

[10]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[11]  François Bouchut,et al.  Relaxation to isentropic gas dynamics for a BGK system with single kinetic entropy , 2002 .

[12]  Francois Dubois,et al.  Une introduction au schma de Boltzmann sur rseau , 2007 .

[13]  François Dubois,et al.  Equivalent partial differential equations of a lattice Boltzmann scheme , 2008, Comput. Math. Appl..

[14]  Feng Chen,et al.  Prandtl number effects in MRT Lattice Boltzmann models for shocked and unshocked compressible fluids , 2011, ArXiv.

[15]  Feng Chen,et al.  Multiple-relaxation-time lattice Boltzmann model for compressible fluids , 2009, 0908.3749.

[16]  François Bouchut,et al.  Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics , 2002 .

[17]  P. Dellar.  Two routes from the Boltzmann equation to compressible flow of polyatomic gases , 2008 .

[18]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[19]  P. Lallemand,et al.  Momentum transfer of a Boltzmann-lattice fluid with boundaries , 2001 .

[20]  Florin Sabac,et al.  The Optimal Convergence Rate of Monotone Finite Difference Methods for Hyperbolic Conservation Laws , 1997 .

[21]  R. Natalini,et al.  Weakly coupled systems of quasilinear hyperbolic equations , 1996, Differential and Integral Equations.

[22]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[23]  R. Natalini,et al.  Convergence of relaxation schemes for conservation laws , 1996 .

[24]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

[25]  Marica Pelanti,et al.  A Relaxation Method for Modeling Two-Phase Shallow Granular Flows , 2008 .

[26]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[27]  B. Shizgal,et al.  Generalized Lattice-Boltzmann Equations , 1994 .