On the well-balance property of Roe?s method for nonconservative hyperbolic systems

This paper is concerned with the numerical approximations of Cauchy problems for one- dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well- balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by (Toumi, J. Comp. Phys. 102 (1992) 360-373). Next, this general theory is applied to obtain well- balanced schemes for solving coupled systems of conservation laws with source terms. Finally, we focus on applications to shallow water systems: the numerical schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by (Bermudez and Vazquez-Cendon, Comput. Fluids 23 (1994) 1049-1071); in the case of two layer flows, they are compared with the numerical scheme presented by (Castro, Mac´ias and Pares, ESAIM: M2AN 35 (2001) 107-127).

[1]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

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

[3]  Enrique D. Fernández Nieto,et al.  An Entropy-correction free solver for non-homogeneous Shallow Water equations , 2003 .

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

[5]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[6]  P. García-Navarro,et al.  On numerical treatment of the source terms in the shallow water equations , 2000 .

[7]  Enrique D. Fernández-Nieto.  Aproximación numérica de leyes de consevación hiperbólicas no homogéneas: aplicación a las ecuaciones de aguas someras , 2003 .

[8]  Andrew C. Fowler,et al.  Mathematical Models in the Applied Sciences , 1997 .

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

[10]  A. Noussair,et al.  Microscopic profiles of shock waves and ambiguities in multiplications of distributions , 1989 .

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

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

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

[14]  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 .

[15]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[16]  E. F. Toro,et al.  Model Hyperbolic Systems with Source Terms: Exact and Numerical Solutions , 2001 .

[17]  Carlos Parés,et al.  Numerical simulation of two-layer shallow water flows through channels with irregular geometry , 2004 .

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

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

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

[21]  Philippe Le Floch,et al.  Propagating phase boundaries: Formulation of the problem and existence via the Glimm method , 1993 .

[22]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[23]  Gerald Warnecke,et al.  On the Solution to The Riemann Problem for the Compressible Duct Flow , 2004, SIAM J. Appl. Math..

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

[25]  Roger Grimshaw,et al.  Water Waves , 2021, Mathematics of Wave Propagation.

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

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

[28]  T. C. Rebollo,et al.  A flux‐splitting solver for shallow water equations with source terms , 2003 .

[29]  M. Vázquez-Cendón.  Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry , 1999 .

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

[31]  P. Goatin,et al.  The Riemann problem for a class of resonant hyperbolic systems of balance laws , 2004 .

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

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

[34]  María Elena Vázquez Cendón,et al.  Estudio de esquemas descentrados para su aplicación a las leyes de conservación hiperbólicas con términos fuente , 1994 .

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

[36]  R. LeVeque.  Approximate Riemann Solvers , 1992 .

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