An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

TL;DR

This paper introduces a high-order, positivity-preserving finite volume WENO method combined with mPDeC time integration for the shallow water equations, ensuring unconditionally positive water heights and demonstrating strong numerical performance.

Contribution

It presents a novel high-order well-balanced scheme that guarantees positivity of water height using mPDeC, with minimal modifications to classical WENO methods.

Findings

Unconditionally positivity-preserving water height in simulations

High-order accuracy demonstrated with fifth-order method

Good performance and verification of theoretical properties

Abstract

In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.