A New Family of Exponential-Based High-Order DGTD Methods for Modeling 3-D Transient Multiscale Electromagnetic Problems

Abstract

A new family of exponential-based time integration methods are proposed for the time-domain Maxwell’s equations discretized by a high-order discontinuous Galerkin (DG) scheme formulated on locally refined unstructured meshes. These methods, which are developed from the Lawson method, remove the stiffness on the time explicit integration of the semidiscrete operator associated with the fine part of the mesh, and allow for the use of high-order time explicit scheme for the coarse part operator. They combine excellent stability properties with the ability to obtain very accurate solutions even for very large time step sizes. Here, the explicit time integration of the Lawson-transformed semidiscrete system relies on a low-storage Runge-Kutta (LSRK) scheme, leading to a combined Lawson-LSRK scheme. In addition, efficient techniques are presented to further improve the efficiency of this exponential-based time integration. For the efficient calculation of matrix exponential, we employ the Krylov subspace method. Numerical experiments are presented to assess the stability, verify the accuracy, and numerical convergence of the Lawson-LSRK scheme. They also demonstrate that the DG time-domain methods based on the proposed time integration scheme can be much faster than those based on classical fully explicit time stepping schemes, with the same accuracy and moderate memory usage increase on locally refined unstructured meshes, and are thus very promising for modeling 3-D multiscale electromagnetic problems.