High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in (Castro et al., Math. Comput. 75:1103–1134, 2006) to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water systems.

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

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

[3]  Jostein R. Natvig,et al.  Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows , 2006, J. Comput. Phys..

[4]  R. LeVeque,et al.  Balancing Source Terms and Flux Gradientsin High-Resolution Godunov Methods : The Quasi-Steady Wave-Propogation AlgorithmRandall , 1998 .

[5]  H. Joachim Schroll,et al.  A Bi–Hyperbolic Finite Volume Method on Quadrilateral Meshes , 2006, J. Sci. Comput..

[6]  Manuel Jesús Castro Díaz,et al.  High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems , 2006, Math. Comput..

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

[8]  Guido Walz.  Romberg Type Cubature over Arbitrary Triangles , 1997 .

[9]  Antonio Marquina,et al.  LOCAL PIECEWISEHYPERBOLIC RECONSTRUCTIONOFNUMERICAL FLUXES FORNONLINEARSCALARCONSERVATION LAWS , 1994 .

[10]  Eleuterio F. Toro,et al.  MUSTA fluxes for systems of conservation laws , 2006, J. Comput. Phys..

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

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

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

[14]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

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

[16]  D. Farmer,et al.  Maximal two-layer exchange through a contraction with barotropic net flow , 1986, Journal of Fluid Mechanics.

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

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

[19]  Susana Serna,et al.  A Class of Extended Limiters Applied to Piecewise Hyperbolic Methods , 2006, SIAM J. Sci. Comput..

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

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

[22]  David M. Farmer,et al.  Maximal two-layer exchange over a sill and through the combination of a sill and contraction with barotropic flow , 1986, Journal of Fluid Mechanics.

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

[24]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[25]  P. Floch.  Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form , 1989 .

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

[27]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[28]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[29]  Tai-Ping Liu,et al.  Existence theory for nonlinear hyperbolic systems in nonconservative form , 1993 .

[30]  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..

[31]  Manuel Jesús Castro Díaz,et al.  Available Online at Www.sciencedirect.com Mathematical and So,snos ~__d,~ot" Computer Modelling the Numerical Treatment of Wet/dry Fronts in Shallow Flows: Application to One-layer and Two-layer Systems , 2022 .

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

[33]  W. Thacker.  Some exact solutions to the nonlinear shallow-water wave equations , 1981, Journal of Fluid Mechanics.

[34]  Yulong Xing,et al.  High order finite difference WENO schemes with the exact conservation property for the shallow water equations , 2005 .