A Comparative Study of Iterative Riemann Solvers for the Shallow Water and Euler Equations

TL;DR

This paper evaluates the efficiency and accuracy of iterative Riemann solvers for shallow water and Euler equations, showing they can approach the speed of approximate methods while providing exact solutions.

Contribution

It demonstrates that robust iterative solvers, with specific modifications, can efficiently compute exact solutions for hyperbolic systems like shallow water and Euler equations.

Findings

Newton's method converges quickly for shallow water equations.

A combination of Ostrowski and Newton iterations accelerates convergence for Euler equations.

Iterative solvers are within a factor of two in speed compared to approximate solvers.

Abstract

The Riemann problem for first-order hyperbolic systems of partial differential equations is of fundamental importance for both theoretical and numerical purposes. Many approximate solvers have been developed for such systems; exact solution algorithms have received less attention because computation of the exact solution typically requires iterative solution of algebraic equations. Iterative algorithms may be less computationally efficient or might fail to converge in some cases. We investigate the achievable efficiency of robust iterative Riemann solvers for relatively simple systems, focusing on the shallow water and Euler equations. We consider a range of initial guesses and iterative schemes applied to an ensemble of test Riemann problems. For the shallow water equations, we find that Newton's method with a simple modification converges quickly and reliably. For the Euler equations we obtain similar results; however, when the required precision is high, a combination of Ostrowski and Newton iterations converges faster. These solvers are slower than standard approximate solvers like Roe and HLLE, but come within a factor of two in speed. We also provide a preliminary comparison of the accuracy of a finite volume discretization using an exact solver versus standard approximate solvers.