A simple Riemann solver and high-order Godunov schemes for hyperbolic systems of conservation laws

A simple approximate Riemann solver for hyperbolic systems of conservation laws is developed for its use in Godunov schemes. The solver is based on characteristic formulations and is illustrated through Euler and ideal magnetohydrodynamical (MHD) equations. The procedure of a high-order Godunov scheme incorporated with the Riemann solver for one-dimensional hyperbolic systems of conservation laws is described in detail. The correctness of the scheme is shown by comparison with the piecewise parabolic method for Euler equations and by comparison with exact solutions of Riemann problems for ideal MHD equations. The robustness of the scheme is demonstrated through numerical examples involving more than one strong shock at the same time. It is shown that the scheme offers the principle advantages of Godunov schemes: robust operation in the presence of strong waves, thin shock fronts, thin contact and slip surface discontinuities