A comparison between bottom-discontinuity numerical treatments in the DG framework

TL;DR

This paper compares five numerical methods within a third-order discontinuous Galerkin framework for simulating free-surface shallow flows over bottom steps, focusing on how they handle bottom discontinuities and preserve steady states.

Contribution

It introduces a comprehensive comparison of existing and new approaches for bottom discontinuity treatment in DG schemes, emphasizing steady state preservation in shallow flow simulations.

Findings

All methods achieve third-order accuracy in unsteady flow simulations.

Certain approaches better preserve steady states, especially in the presence of bottom steps.

The new approach shows promising results for moving-water steady states.

Abstract

In this work, using a unified framework consisting of third-order accurate discontinuous Galerkin schemes, we perform a comparison between five different numerical approaches to the free-surface shallow flow simulation on bottom steps. Together with the study of the overall impact that such techniques have on the numerical models, we highlight the role that the treatment of bottom discontinuities plays in the preservation of specific asymptotic conditions. In particular, we consider three widespread approaches that perform well if the motionless steady state has to be preserved and two approaches (one previously conceived by the first two authors and one original) which are also promising for the preservation of a moving-water steady state. Several one-dimensional test cases are used to verify the third-order accuracy of the models in simulating an unsteady flow, the behavior of the models for a quiescent flow in the cases of both continuous and discontinuous bottom, and the good resolution properties of the schemes. Moreover, specific test cases are introduced to show the behavior of the different approaches when a bottom step interacts with both steady and unsteady moving flows.