An ADER discontinuous Galerkin method with local time-stepping for transient electromagnetics

This paper is concerned with a novel numerical method of arbitrary order of accuracy in both space and time for transient electromagnetic problems. The scheme is based on the combination of discontinuous Galerkin (DG) spectral elements for spatial discretization with a high order Taylor type integrator (TTI) for time-stepping. With the electromagnetic degrees of freedom for discontinuous elements expanded into Taylor series, a one-step single-stage TTI is obtained by replacing the time derivatives with space derivatives. Since the time-dependent Maxwell's equations semi-discretized in space are used repeatedly for this substitution, no tedious analytical formula is involved. The new TTI needs only one extra memory unit for each unknown. The DG method along with vector spectral elements is employed to treat the spatial discretization. A mortar face element is introduced to buffer the communicative information between any two adjacent grids, which enables the whole system to be updated iteratively in a highly efficient and memory-saving way. A local time-stepping technique accompanying the TTI is proposed to enhance the speed of time-marching on multi-scale meshes. To demonstrate robustness and advantages of the scheme, several examples meshed with curved elements are simulated. The present method is particularly suitable for solving complex electrically large problems with requirements for high accuracy.