A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms
type: Item
instance of: scholarly article
title: A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms
zbMATH Open document ID: 0963.65090
DOI: 10.1016/S0898-1221(00)00093-6
publication date: 2001-05-31T00:00:00Z
review text: Well balanced flux vector splitting schemes are developed for hyperbolic conservation laws with source term: \(u_t+F(u)_x=G(u).\) The approach for a building of well balanced schemes originates from \textit{J. M. Greenberg} and \textit{A. Y. Leroux} [SIAM J. Numer. Anal. 33, 1-16 (1996; Zbl 0876.65064)]. First of all a nonlinear scalar conservation law is considered where the source term \(G(u)\) satisfies \(G(0)=0\) and \(uG(u)<0\) for \(|u|\) large enough. The main idea to design a scheme is the following: A new dependent variable \(a\) is introduced, \(a(x)=x\), \(a_t=0\) and the source term is converted into equivalent form \(G(u)a_x\); for resulting the nonconservative system without source term the scheme is designed by means of solving appropriate Riemann problems at cell interfaces. This approach is formulated for strictly hyperbolic systems of conservation laws as well. The approach is extended for the flux vector splitting method by means of considering two suitable Riemann problems for hyperbolic systems containing fluxes corresponding to nonpositive and nonnegative characteristic directions in accordance with splitting of flux vector of original system of equations. Well balancing property is proved for developed flux vector splitting schemes. In two space dimension extension of the scheme on rectangular grid is performed by means of suitable directional splitting of an equation and a source, term thus arriving to 1D case for each coordinate directions. For shallow water equations and 2D simplified two phase flow equations, well balanced flux vector splitting schemes are given. In general, e.g. in case of shallow water equations, the scheme does not take care of mass conservation property that is important for unsteady flow computations. Numerical results presented in the paper show that well balanced schemes can accurately compute steady state solutions.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 76M20
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 1471226
zbMATH Keywords: shallow water equations
zbMATH Keywords: Riemann problems
zbMATH Keywords: nonconservative product
Wikidata QID: Q115023117
author: Laurent Gosse
reviewed by: Ramaz Botchorishvili
MaRDI profile type: Publication
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full work available at URL: s0898-1221(00)00093-6
OpenAlex ID: W2084025336
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1471226