Convergence of second-order, entropy stable methods for multi-dimensional conservation laws

High-order accurate, $\textit{entropy stable}$ numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space dimensions. In this paper we show how the entropy stability of one such method yields a (weak) bound on oscillations, and using compensated compactness we prove convergence to a weak solution satisfying at least one entropy condition.

[1]  E. Panov.  Erratum to: Existence and Strong Pre-compactness Properties for Entropy Solutions of a First-Order Quasilinear Equation with Discontinuous Flux , 2010 .

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

[3]  Philippe G. LeFloch,et al.  Fully Discrete, Entropy Conservative Schemes of ArbitraryOrder , 2002, SIAM J. Numer. Anal..

[4]  Christian Rohde,et al.  Error Estimates for Finite Volume Approximations of Classical Solutions for Nonlinear Systems of Hyperbolic Balance Laws , 2006, SIAM J. Numer. Anal..

[5]  Mathematisches Forschungsinstitut Oberwolfach,et al.  Hyperbolic Conservation Laws , 2004 .

[6]  Eitan Tadmor,et al.  ENO Reconstruction and ENO Interpolation Are Stable , 2011, Found. Comput. Math..

[7]  S. Osher.  Riemann Solvers, the Entropy Condition, and Difference , 1984 .

[8]  S. Osher,et al.  Some results on uniformly high-order accurate essentially nonoscillatory schemes , 1986 .

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

[10]  James M. Hyman,et al.  On Finite-Difference Approximations and Entropy Conditions for Shocks , 2015 .

[11]  Eitan Tadmor,et al.  Numerical Viscosity and the Entropy Condition for Conservative Difference Schemes , 1984 .

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

[13]  Luc Tartar,et al.  Compensated compactness and applications to partial differential equations , 1979 .

[14]  Rémi Abgrall,et al.  Handbook of Numerical Methods for Hyperbolic Problems : Basic and Fundamental Issues , 2016 .

[15]  Ulrik Skre Fjordholm,et al.  High-order accurate entropy stable numercial schemes for hyperbolic conservation laws , 2013 .

[16]  Ulrik Skre Fjordholm,et al.  High-order accurate, fully discrete entropy stable schemes for scalar conservation laws , 2016 .

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

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

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

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

[21]  Eitan Tadmor,et al.  Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems , 2003, Acta Numerica.

[22]  U. S. Fjordholm.  Stability Properties of the ENO Method , 2016, 1609.04178.

[23]  Chi-Wang Shu,et al.  Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws , 2017, J. Comput. Phys..