Bounds for Wave Speeds in the Riemann Problem: Direct Theoretical Estimates

TL;DR

This paper develops direct, non-iterative bounds for the fastest wave speeds in the Riemann problem across three hyperbolic systems, improving the reliability of numerical methods by providing accurate speed estimates.

Contribution

It introduces new theoretical bounds for wave speeds in the Riemann problem that are validated against exact solutions and outperform previous estimates.

Findings

Derived bounds are confirmed to be correct, bounding the true wave speeds from below and above.

Previous estimates often do not bound the true wave speeds, highlighting the need for improved bounds.

The bounds are applicable to various hyperbolic systems, aiding in stable numerical simulations.

Abstract

In this paper we provide bound estimates for the two fastest wave speeds emerging from the solution of the Riemann problem for three well-known hyperbolic systems, namely the Euler equations of gas dynamics, the shallow water equations and the blood flow equations for arteries. Several approaches are presented, all being direct, that is non-iterative. The resulting bounds range from crude but simple estimates to accurate but sophisticated estimates that make limited use of information from the solution of the Riemann problem. Through a carefully chosen suite of test problems we asses our wave speed estimates against exact solutions and against previously proposed wave speed estimates. The results confirm that the derived theoretical bounds are actually so, from below and above, for minimal and maximal wave speeds respectively. The results also show that popular previously proposed estimates do not bound the true wave speeds in general. Applications in mind, but not pursued here, include (i) reliable implementation of the Courant condition to determine a stable time step in all explicit methods for hyperbolic equations; (ii) use in local time stepping algorithms and (iii) construction of HLL-type numerical fluxes for hyperbolic equations.