Riemann Solver for Hyperbolic Equations with Discontinuous Coefficients: A Mathematical Proof of the Constant State Formula

Abstract

In Godunov numerical methods type used in many industrial and scientific numerical simulations including; fluid dynamics, electromagnetic, electro-hydrodynamic problems, a Riemann problem needs to be solved to estimate fluxes. The exact solution is generally not possible to obtain, but good approximations are available,  Roe and HLLC Riemann solvers are among the most popular. However, all these solvers assume that the acoustic waves speeds are continuous by considering some averaging. In a previous work the effect of such averaging is demonstrated to be significant for some applications leading to a wrong solution. A Riemann solvers is proposed taking into account the discontinuity of the acoustic waves speeds. The case that shows discrepancy comparing to the averaged  solvers is the one with an acoustic waves speeds having a negative left value and a positive right value. In this case a constant state appears and a formula of the constant state is proposed. A numerical, and a particular exact solution based on a regularization technique are provided to demonstrate the validity of the formula. However, and due to the important impact of this case on   Godunov type schemes, a mathematical proof is necessary. In this paper the formula of the constant state is proved, the proof is based on the generalized functions algebra theory .