Approximate Osher–Solomon schemes for hyperbolic systems

This paper is concerned with a new kind of Riemann solvers for hyperbolic systems, which can be applied both in the conservative and nonconservative cases. In particular, the proposed schemes constitute a simple version of the classical Osher–Solomon Riemann solver, and extend in some sense the schemes proposed in Dumbser and Toro (2011) [19,20]. The viscosity matrix of the numerical flux is constructed as a linear combination of functional evaluations of the Jacobian of the flux at several quadrature points. Some families of functions have been proposed to this end: Chebyshev polynomials and rational-type functions. Our schemes have been tested with different initial value Riemann problems for ideal gas dynamics, magnetohydrodynamics and multilayer shallow water equations. The numerical tests indicate that the proposed schemes are robust, stable and accurate with a satisfactory time step restriction, and provide an efficient alternative for approximating time-dependent solutions in which the spectral decomposition is computationally expensive.