A Conservative Eulerian-Lagrangian Runge-Kutta Discontinuous Galerkin Method for Linear Hyperbolic System with Large Time Stepping

We propose an Eulerian-Lagrangian (EL) Runge-Kutta (RK) discontinuous Galerkin (DG) method for a linear hyperbolic system. The method is designed based on the ELDG method for transport problems (J Comput Phys 446:110,632, 2021), which tracks solutions along approximations to characteristics in the DG framework, allowing extra large time stepping sizes with stability with respect to the classical RKDG method. Considering each characteristic family, a straightforward application of ELDG for the hyperbolic system 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 conservation could not be guaranteed in a general setting. In this paper, we formulate a 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 for 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 conservative property.