A Generalized Eulerian-Lagrangian Discontinuous Galerkin Method for Transport Problems

We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerkin (DG) method. The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in [arXiv preprint arXiv: 2002.02930 (2020)], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping size with stability. The newly proposed GEL DG method in this paper is motivated for solving linear hyperbolic systems with variable coefficients, where the velocity field for adjoint problems of the test functions is frozen to constant. In this paper, in a simplified scalar setting, we propose the GEL DG methodology by freezing the velocity field of adjoint problems, and by formulating the semi-discrete scheme over the space-time region partitioned by linear lines approximating characteristics. The fully-discrete schemes are obtained by method-of-lines Runge-Kutta methods. We further design flux limiters for the schemes to satisfy the discrete geometric conservation law (DGCL) and maximum principle preserving (MPP) properties. Numerical results on 1D and 2D linear transport problems are presented to demonstrate great properties of the GEL DG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and satisfaction of DGCL and MPP properties.