A Hybrid Riemann Solver for Large Hyperbolic Systems of Conservation Laws

TL;DR

This paper introduces HLLXω, a family of simple, efficient Riemann solvers for large hyperbolic systems that avoid eigensystem computations while capturing all waves with less dissipation.

Contribution

The paper presents a new family of Riemann solvers, HLLXω, that efficiently approximate wave propagation without eigensystem calculations, improving accuracy over existing methods like HLL and FORCE.

Findings

HLLXω reproduces all waves with less dissipation.

The method is computationally efficient for large systems.

It requires only estimates of wave speeds, not eigensystems.

Abstract

We are interested in the numerical solution of large systems of hyperbolic conservation laws or systems in which the characteristic decomposition is expensive to compute. Solving such equations using finite volumes or Discontinuous Galerkin requires a numerical flux function which solves local Riemann problems at cell interfaces. There are various methods to express the numerical flux function. On the one end, there is the robust but very diffusive Lax-Friedrichs solver; on the other end the upwind Godunov solver which respects all resulting waves. The drawback of the latter method is the costly computation of the eigensystem. This work presents a family of simple first order Riemann solvers, named HLLX$\omega$, which avoid solving the eigensystem. The new method reproduces all waves of the system with less dissipation than other solvers with similar input and effort, such as HLL and FORCE. The family of Riemann solvers can be seen as an extension or generalization of the methods introduced by Degond et al. \cite{DegondPeyrardRussoVilledieu1999}. We only require the same number of input values as HLL, namely the globally fastest wave speeds in both directions, or an estimate of the speeds. Thus, the new family of Riemann solvers is particularly efficient for large systems of conservation laws when the spectral decomposition is expensive to compute or no explicit expression for the eigensystem is available.