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A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. (English) Zbl 0963.65090
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 J. M. Greenberg and 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.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76M20 Finite difference methods applied to problems in fluid mechanics 35L65 Hyperbolic conservation laws
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