Reliability of first order numerical schemes for solving shallow water system over abrupt topography

TL;DR

This paper evaluates the reliability of first order well-balanced numerical schemes for shallow water equations over abrupt topography, highlighting the importance of spatial resolution and proposing modifications to improve accuracy near discontinuities.

Contribution

It compares different schemes, analyzes their error dependence on spatial resolution, and introduces a modification to the hydrostatic reconstruction technique for better handling of large bottom discontinuities.

Findings

Solution accuracy depends on space step size.

Large bottom discontinuities can be neglected without modification.

Proposed modification improves handling of abrupt topography.

Abstract

We compare some first order well-balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography. We show that the space step required to obtain a prescribed error depends on the method. Moreover, the solutions given by the numerical scheme can be significantly different if not enough space resolution is used. We shall pay special attention to the well-known hydrostatic reconstruction technique where it is shown that large bottom discontinuities may be neglected and a modification is proposed to avoid this problem.