A mass conservative Eulerian-Lagrangian Runge-Kutta discontinuous Galerkin method for wave equations with large time stepping

TL;DR

This paper introduces a mass conservative Eulerian-Lagrangian Runge-Kutta discontinuous Galerkin method for wave equations that enables large time steps while maintaining stability and accuracy.

Contribution

It develops a novel mass conservative semi-discrete ELDG method for wave equations, allowing large time steps with guaranteed mass conservation.

Findings

High order spatial and temporal accuracy demonstrated

Stable with extra large time stepping sizes

Mass conservation property achieved

Abstract

We propose an Eulerian-Lagrangian (EL) Runge-Kutta (RK) discontinuous Galerkin (DG) method for wave equations. The method is designed based on the ELDG method for transport problems [J. Comput. Phy. 446: 110632, 2021.], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping sizes with stability. The wave equation can be written as a first order hyperbolic system. Considering each characteristic family, a straightforward application of ELDG will be to transform to the characteristic variables, evolve them on associated characteristic related space-time regions, and transform them back to the original variables. However, the mass conservation could not be guaranteed in a general setting. In this paper, we formulate a mass conservative semi-discrete ELDG method by decomposing each variable into two parts, each of them associated with a different characteristic family. As a result, four different quantities are evolved in EL fashion and recombined to update the solution. The fully discrete scheme is formulated by using method-of-lines RK methods, with intermediate RK solutions updated on the background mesh. Numerical results on 1D and 2D wave equations are presented to demonstrate the performance of the proposed ELDG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and mass conservative property.