To Split or Not to Split, That Is the Question in Some Shallow Water Equations

TL;DR

This paper investigates the effectiveness of time splitting schemes for solving shallow water equations, emphasizing the importance of well-balanced properties to accurately handle source terms and shock interactions.

Contribution

It provides a detailed analysis of when approximate versus exact C-property verification is necessary for these schemes, supported by numerical tests.

Findings

Well-balanced schemes are crucial for accurate shallow water simulations.

Approximate C-property suffices in some cases, while exact C-property is needed in others.

Numerical tests validate the theoretical analysis.

Abstract

In this paper we analyze the use of time splitting techniques for solving shallow water equation. We discuss some properties that these schemes should satisfy so that interactions between the source term and the shock waves are controlled. This paper shows that these schemes must be well balanced in the meaning expressed by Greenberg and Leroux [5]. More specifically, we analyze in what cases it is enough to verify an Approximate C-property and in which cases it is required to verify an Exact C-property (see [1], [2]). We also include some numerical tests in order to justify our reasoning.