An Eulerian-Lagrangian discontinuous Galerkin method for wave equations

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 proposed in [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 straight forward application of ELDG will be to project to the characteristic variables, evolve them on associated space-time regions, and project them back to the original variables. However, mass conservation could not be guaranteed in a general setting. In this talk, we formulated a mass conservative semi-discrete ELDG method by decomposing each variable into two parts, each of them associated with different characteristic families. 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.