A Well Balanced Reconstruction with Bounded Velocities and Low-Oscillation Slow Shocks for the Shallow Water Equations

TL;DR

This paper evaluates and improves a well balanced reconstruction scheme for the shallow water equations, enabling accurate steady state simulation and resolving slow shocks without losing stability or accuracy.

Contribution

It demonstrates the effectiveness of Skevington's scheme and introduces a modification to handle slow shocks and dam break problems more robustly.

Findings

Accurately resolves steady and near steady states.

Enables resolution of slow-moving shocks and dam breaks.

Maintains well balanced property with modifications.

Abstract

Many numerical schemes for hyperbolic systems require a piecewise polynomial reconstruction of the cell averaged values, and to simulate perturbed steady states accurately we require a so called 'well balanced' reconstruction scheme. For the shallow water system this involves reconstructing in surface elevation, to which modifications must be made as the fluid depth becomes small to ensure positivity. We investigate the scheme proposed in Skevington (2021) though numerical experiments, demonstrating its ability to resolve steady and near steady states at high accuracy. We also present a modification to the scheme which enables the resolution of slowly moving shocks and dam break problems without compromising the well balanced property.