Runge-Kutta discontinuous local evolution Galerkin methods for the shallow water equations on the cubed-sphere

TL;DR

This paper introduces high-order Runge-Kutta discontinuous local evolution Galerkin methods on the cubed-sphere grid for the shallow water equations, emphasizing multi-dimensional operators over traditional splitting techniques.

Contribution

It presents a novel multi-dimensional approach for solving SWEs on the cubed-sphere, avoiding dimensional splitting and one-dimensional Riemann problems.

Findings

Demonstrates high accuracy of RKDLEG methods.

Shows improved performance over traditional Runge-Kutta discontinuous Galerkin methods.

Validates methods through numerical experiments.

Abstract

The paper develops high order accurate Runge-Kutta discontinuous local evolution Galerkin (RKDLEG) methods on the cubed-sphere grid for the shallow water equations (SWEs). Instead of using the dimensional splitting method or solving one-dimensional Riemann problem in the direction normal to the cell interface, the RKDLEG methods are built on genuinely multi-dimensional approximate local evolution operator of the locally linearized SWEs on a sphere by considering all bicharacteristic directions. Several numerical experiments are conducted to demonstrate the accuracy and performance of our RKDLEG methods, in comparison to the Runge-Kutta discontinuous Galerkin method with Godunov's flux etc.