Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws

Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the underlying viscosity mechanism and approximate the physically relevant solution. Numerical experiments illustrating the robust performance of these schemes are presented.

[1]  S. Bianchini,et al.  The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation , 2006, math/0605575.

[2]  A. Bressan,et al.  Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems , 2001, math/0111321.

[3]  Denis Serre,et al.  Discrete Shock Profiles: Existence and Stability , 2007 .

[4]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[5]  M. Sablé-Tougeron,et al.  Méthode de Glimm et problème mixte , 1993 .

[6]  S. Bianchini.  On the Riemann Problem for Non-Conservative Hyperbolic Systems , 2003 .

[7]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[8]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[9]  L. Spinolo.  ov 2 00 5 Vanishing viscosity solutions of a 2 × 2 triangular hyperbolic system with Dirichlet conditions on two boundaries , 2005 .

[10]  Tai-Ping Liu,et al.  The Riemann problem for general systems of conservation laws , 1975 .

[11]  Laura V. Spinolo.  Vanishing viscosity solutions of a $2 \times 2$ triangular hyperbolic system with Dirichlet conditions on two boundaries , 2005 .

[13]  Shuichi Kawashima,et al.  On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws , 1988 .

[14]  Siddhartha Mishra,et al.  Accurate numerical discretizations of non-conservative hyperbolic systems , 2012 .

[15]  J. Goodman.  Initial boundary value problems for hyperbolic systems of conservation laws , 1982 .

[16]  Philippe G. LeFloch,et al.  Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues , 2010, 1002.2950.

[17]  Eitan Tadmor,et al.  Arbitrarily High-order Accurate Entropy Stable Essentially Nonoscillatory Schemes for Systems of Conservation Laws , 2012, SIAM J. Numer. Anal..

[18]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[19]  I. N. Sneddon,et al.  Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves , 1999 .

[20]  M. Gisclon,et al.  Etude des conditions aux limites pour des systèmes strictement hyperboliques, via l'approximation parabolique , 1994 .

[21]  Tai-Ping Liu,et al.  The entropy condition and the admissibility of shocks , 1976 .

[22]  D. Amadori.  Initial-boundary value problems for nonlinear systems of conservation laws , 1997 .

[23]  D. Serre.  Systems of conservation laws , 1999 .

[24]  Manuel Jesús Castro Díaz,et al.  Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes , 2008, J. Comput. Phys..

[25]  Siddhartha Mishra,et al.  OPTIMAL ENTROPY SOLUTIONS FOR CONSERVATION LAWS WITH DISCONTINUOUS FLUX-FUNCTIONS , 2005 .

[26]  A. Bressan.  Hyperbolic Systems of Conservation Laws , 1999 .

[27]  F. Ancona,et al.  Vanishing Viscosity Solutions of Hyperbolic Systems of Conservation Laws with Boundary , 2006 .

[28]  P. Lax.  Hyperbolic systems of conservation laws II , 1957 .

[29]  Philip L. Roe,et al.  Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks , 2009, J. Comput. Phys..

[30]  Philippe G. LeFloch,et al.  Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws , 1999 .

[31]  Eitan Tadmor,et al.  The numerical viscosity of entropy stable schemes for systems of conservation laws. I , 1987 .

[32]  P. Floch,et al.  Boundary conditions for nonlinear hyperbolic systems of conservation laws , 1988 .

[33]  Laura V. Spinolo,et al.  Vanishing viscosity solution of a 2x2 triangular hyperbolic system with Dirichlet conditions on towo boundaries , 2007 .