High-order explicit time-integrators for discontinuous Galerkin discretizations of the Maxwell equations
Résumé
We investigate the practical implementation of a high-order explicit time-stepping method based on polynomial approximations, for possible application to large-scale problems in electromagnetics. After the spatial discretization by a high-order discontinuous Galerkin method, we obtain a linear system of differential equations of the form, $\partial_t Y(t) = H Y(t) + S(t)$, where H is a matrix containing the spatial derivatives and t is the time variable. The formal solution can be written in terms of the matrix exponential, exp(tH), acting on some vectors. We introduce a general family of time-integrators based on the approximation of exp(tH) by Jacobi polynomial expansions. We discuss the efficient implementation of this technique, and based on some test problems, we compare the virtues and shortcomings of the algorithm. We also demonstrate how these schemes provide an efficient alternative to standard explicit integrators for computing solutions over long time intervals.