Influence of Total-Variation-Diminishing Slope Limiting on Local Discontinuous Galerkin Solutions of the Shallow Water Equations

Abstract

Finite volume (FV) slope limiting is essential to stabilize discontinuous Galerkin (DG) solutions despite a number of side effects such as local loss of accuracy and increased run-time cost. These side effects have been experienced with DG solutions to the homogeneous system of conservation laws and are usually accepted as long as they do not affect the reliability of the numerical predictions and provide a better stability property. They have also led to the concept of localizing the slope-limiting process. When a model is applied to simulate the flow problems that necessitate the solution of the inhomogeneous shallow water equations (SWEs) with/without flooding and drying, the slope-limiting routine can have extra adverse effects on the local DG framework. These effects, causes, and implications on DG solutions to the SWEs have not yet received adequate attention, and a full investigation is therefore needed. With the aim of improving a second-order Runge-Kutta discontinuous Galerkin (RKDG2) SWE solver, the effects of FV slope limiting on the RKDG2 flow solutions were analyzed. Three versions of the RKDG2 scheme were configured: the first and second versions were respectively associated with the local limiting (LL) and global limiting (GL) processes, which both use limiter-free linear front tracking (LFT) to trace a wet/dry front, and the third was implemented with the local limiting process but with constant front-tracking (CFT) of the wet/dry front. Selected hypothetical and practical hydraulic tests were investigated to demonstrate the effects of different limiting processes and front-tracking techniques.