Entropy Conservative and Entropy Stable Schemes for Nonconservative Hyperbolic Systems

The vanishing viscosity limit of nonconservative hyperbolic systems depends heavily on the specific form of the viscosity. Numerical approximations, such as the path consistent schemes of [C. Pares, SIAM J. Numer. Anal., 41 (2007), pp. 169--185], may not converge to the physically relevant solutions of the system. We construct entropy stable path consistent (ESPC) schemes to approximate nonconservative hyperbolic systems by combining entropy conservative discretizations with numerical diffusion operators that are based on the underlying viscous operator. Numerical experiments for the coupled Burgers system and the two-layer shallow water equations demonstrating the robustness of ESPC schemes are presented.

[1]  Carlos Parés Madroñal,et al.  Numerical methods for nonconservative hyperbolic systems: a theoretical framework , 2006, SIAM J. Numer. Anal..

[2]  Ulrik Skre Fjordholm.  Energy Conservative and Stable Schemes for the Two-layer Shallow Water Equations , 2012 .

[3]  Smadar Karni,et al.  Viscous shock profiles and primitive formulations , 1992 .

[4]  Carlos Parés,et al.  A Q-SCHEME FOR A CLASS OF SYSTEMS OF COUPLED CONSERVATION LAWS WITH SOURCE TERM. APPLICATION TO A TWO-LAYER 1-D SHALLOW WATER SYSTEM , 2001 .

[5]  T. Hou,et al.  Why nonconservative schemes converge to wrong solutions: error analysis , 1994 .

[6]  Carlos Parés Madroñal,et al.  On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws , 2011, J. Sci. Comput..

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

[8]  G. D. Maso,et al.  Definition and weak stability of nonconservative products , 1995 .

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

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

[11]  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..

[12]  Carlos Parés,et al.  On the well-balance property of Roe?s method for nonconservative hyperbolic systems , 2004 .

[13]  Rémi Abgrall,et al.  A comment on the computation of non-conservative products , 2010, J. Comput. Phys..

[14]  Enrique D. Fernández-Nieto,et al.  Derivation of a Bilayer Model for Shallow Water Equations with Viscosity. Numerical Validation , 2009 .

[15]  Eitan Tadmor,et al.  Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography , 2011, J. Comput. Phys..

[16]  Alexander Kurganov,et al.  Central-Upwind Schemes for Two-Layer Shallow Water Equations , 2009, SIAM J. Sci. Comput..

[17]  Eitan Tadmor,et al.  Energy Preserving and Energy Stable Schemes for the Shallow Water Equations , 2009 .

[18]  Carlos Parés Madroñal,et al.  On some difficulties of the numerical approximation of nonconservative hyperbolic systems , 2009 .