Numerical methods for nonconservative hyperbolic systems: a theoretical framework

The goal of this paper is to provide a theoretical framework allowing one to extend some general concepts related to the numerical approximation of 1-d conservation laws to the more general case of first order quasi-linear hyperbolic systems. In particular this framework is intended to be useful for the design and analysis of well-balanced numerical schemes for solving balance laws or coupled systems of conservation laws. First, the concept of path-conservative numerical schemes is introduced, which is a generalization of the concept of conservative schemes for systems of conservation laws. Then, we introduce the general definition of approximate Riemann solvers and give the general expression of some well-known families of schemes based on these solvers: Godunov, Roe, and relaxation methods. Finally, the general form of a high order scheme based on a first order path-conservative scheme and a reconstruction operator is presented.

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

[2]  Chiara Simeoni,et al.  Convergence of the Upwind Interface Source Method for Hyperbolic Conservation Laws , 2003 .

[3]  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 .

[4]  Philip L. Roe,et al.  Upwind differencing schemes for hyperbolic conservation laws with source terms , 1987 .

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

[6]  Enrique D. Fernández Nieto,et al.  Asymptotically balanced schemes for non-homogeneous hyperbolic systems – application to the Shallow Water equations , 2004 .

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

[8]  Tao Tang,et al.  Error bounds for fractional step methods for conservation laws with source terms , 1995 .

[9]  J. Colombeau,et al.  Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations , 1989 .

[10]  Arnaud Heibig,et al.  Nonconservative products in bounded variation functions , 1992 .

[11]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[12]  Enrique D. Fernández Nieto,et al.  A family of stable numerical solvers for the shallow water equations with source terms , 2003 .

[13]  Philippe G. LeFloch.  Propagating phase boundaries: formulation of the problem and existence via the Glimm scheme , 2007 .

[14]  A. Tzavaras,et al.  Representation of weak limits and definition of nonconservative products , 1999 .

[15]  Randall J. LeVeque,et al.  Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods , 1998 .

[16]  J. Greenberg,et al.  A well-balanced scheme for the numerical processing of source terms in hyperbolic equations , 1996 .

[17]  Tanmay Vachaspati.  Propagating phase boundaries as sonic horizons , 2003 .

[18]  Pierre-Arnaud Raviart,et al.  A NONCONSERVATIVE HYPERBOLIC SYSTEM MODELING SPRAY DYNAMICS. PART I: SOLUTION OF THE RIEMANN PROBLEM , 1995 .

[19]  Emmanuel Audusse,et al.  A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows , 2004, SIAM J. Sci. Comput..

[20]  Manuel Jesús Castro Díaz,et al.  On Well-Balanced Finite Volume Methods for Nonconservative Nonhomogeneous Hyperbolic Systems , 2007, SIAM J. Sci. Comput..

[21]  Laurent Gosse,et al.  Localization effects and measure source terms in numerical schemes for balance laws , 2002, Math. Comput..

[22]  Laurent Gosse,et al.  A Well-Balanced Scheme Using Non-Conservative Products Designed for Hyperbolic Systems of Conservati , 2001 .

[23]  Kun Xu,et al.  A gas-kinetic scheme for shallow-water equations with source terms , 2004 .

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

[25]  Graziano Guerra.  Well-posedness for a scalar conservation law with singular nonconservative source , 2004 .

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

[27]  L. Gosse.  A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms☆ , 2000 .

[28]  François Alouges,et al.  APPROXIMATE SHOCK CURVES FOR NON-CONSERVATIVE HYPERBOLIC SYSTEMS IN ONE SPACE DIMENSION , 2004 .

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

[30]  J. Greenberg,et al.  Analysis and Approximation of Conservation Laws with Source Terms , 1997 .

[31]  B. Perthame,et al.  A kinetic scheme for the Saint-Venant system¶with a source term , 2001 .

[32]  Florian de Vuyst.  Schémas non-conservatifs et schémas cinétiques pour la simulation numérique d'écoulements hypersoniques non visqueux en déséquilibre thermochimique , 1994 .

[33]  Carlos Parés,et al.  Godunov method for nonconservative hyperbolic systems , 2007 .

[34]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

[35]  E. Toro.  Shock-Capturing Methods for Free-Surface Shallow Flows , 2001 .

[36]  F. James,et al.  One-dimensional transport equations with discontinuous coefficients , 1998 .

[37]  A. I. Vol'pert.  THE SPACES BV AND QUASILINEAR EQUATIONS , 1967 .

[38]  I. Toumi.  A weak formulation of roe's approximate riemann solver , 1992 .

[39]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .