Waves Speed Averaging Impact on Godunov type Schemes for Hyperbolic Equations with Discontinuous Coefficients: The linear scalar case

TL;DR

This paper investigates how averaging wave speeds in Godunov schemes affects solutions for linear scalar hyperbolic equations with discontinuous coefficients, highlighting potential inaccuracies caused by such averaging.

Contribution

It introduces a new regularization-based argument to validate a Riemann solver that accounts for wave speed discontinuities in linear scalar hyperbolic equations.

Findings

A new Riemann solver considering wave speed discontinuities is validated.

Wave speed averaging can lead to incorrect solutions, linked to the distribution product phenomenon.

The derived Godunov scheme demonstrates the impact of wave speed averaging on numerical solutions.

Abstract

This paper deals with the waves speed averaging impact impact on Godunov type schemes for linear scalar hyperbolic equations with discontinuous coefficients. In many numerical schemes of Godunov type used in fluid dynamics, electromagnetic, electro-hydrodynamic problems and so on, usually a Riemann problem needs to be solved to estimate fluxes. The exact solution is generally not possible to obtain, but good approximations are provided in many situations like Roe and HLLC Riemann solvers in fluids. However all these solvers assume that the acoustic waves speed are continuous by considering some averaging. This could unfortunately lead to a wrong solution as we will show in this paper for the linear scalar case. Providing a Riemann solver in the general case of non-linear hyperbolic systems with discontinuous waves speed is a very hard task, therefore in this paper and as a first step, we focus on the linear and scalar case. In a previous work we proposed for such problems a Riemann solution that takes into account the discontinuities of the waves speed, we provided a numerical argument to show the validity of the solution. In this paper, first a new argument using regularization technique is provided to reinforce the validity of the proposed solution. Then, the corresponding Godunov scheme is derived and the effect of waves speed averaging is clearly demonstrated with a clear connection to the distribution product phenomenon.