A new family of exponential-based high order DGTD methods for modelling 3D transient multiscale electromagnetic problems
Résumé
—The accurate and efficient simulation of 3D transient multiscale electromagnetic problems is extremely challenging for conventional numerical methods. Assuming a splitting of the underlying tetrahedral mesh in coarse and fine parts and using the Lawson procedure, we derive a family of exponential-based time integration methods for the time-domain Maxwell's equations discretized by a high order discontinuous Galerkin (DG) scheme formulated on locally refined unstructured meshes. These methods remove the stiffness on the time explicit integration of the semi-discrete operator associated to 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 semi-discrete system relies on a Low-Storage Runge-Kutta (LSRK) scheme, leading to a combined Lawson-LSRK scheme. In addition, efficient techniques are also 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 DGTD 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 modelling three-dimensional multiscale electromagnetic problems.
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