Analytical implementation of Roe solver for two-layer shallow water equations with accurate treatment for loss of hyperbolicity

TL;DR

This paper introduces an analytical Roe solver for two-layer shallow water equations that improves computational efficiency and maintains hyperbolicity, outperforming traditional numerical eigensolvers in speed while preserving accuracy.

Contribution

The paper presents an analytical solution to the characteristic quartic for the Roe scheme, enabling faster computation and better hyperbolicity management in two-layer shallow water models.

Findings

A-Roe scheme is as accurate as the traditional Roe scheme.

A-Roe is significantly faster, comparable to GFORCE and IFCP schemes.

Effective iterative method for hyperbolicity loss mitigation.

Abstract

A new implementation of the Roe scheme for solving two-layer shallow-water equations is presented in this paper. The proposed A-Roe scheme is based on the analytical solution to the characteristic quartic of the flux matrix, which is an efficient alternative to a numerical eigensolver. Additionally, an accurate method for maintaining the hyperbolic character of the governing system is proposed. The efficiency of the quartic closed-form solver is examined and compared to numerical eigensolvers. Furthermore, the accuracy and computational speed of the A-Roe scheme is compared to the Roe, Lax-Friedrichs, GFORCE, PVM, and IFCP schemes. Finally, numerical tests are presented to evaluate the efficiency of the iterative treatment for the hyperbolicity loss. The proposed A-Roe scheme is as accurate as the Roe scheme, but much faster, with computational speeds closer to the GFORCE and IFCP scheme.