A well-balanced Active Flux method for the shallow water equations with wetting and drying

TL;DR

This paper extends the Active Flux numerical method to the nonlinear shallow water equations, achieving a well-balanced, positivity-preserving scheme capable of handling dry states with high accuracy and stability.

Contribution

It introduces the first well-balanced, positivity-preserving Active Flux method for shallow water equations with wetting and drying, using a novel high-order evolution operator.

Findings

Method performs well on test problems with shocks

Achieves third order accuracy in complex scenarios

Successfully handles dry states and bottom topography

Abstract

Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.