An Efficient High-Order Time Integration Method for Spectral-Element Discontinuous Galerkin Simulations in Electromagnetics

We investigate efficient algorithms and a practical implementation of an explicit-type high-order timestepping method based on Krylov subspace approximations, for possible application to large-scale engineering problems in electromagnetics. We consider a semi-discrete form of the Maxwell’s equations resulting from a high-order spectral-element discontinuous Galerkin discretization in space whose solution can be expressed analytically by a large matrix exponential of dimension $kappa times kappa $ . We project the matrix exponential into a small Krylov subspace by the Arnoldi process based on the modified Gram–Schmidt algorithm and perform a matrix exponential operation with a much smaller matrix of dimension $mtimes m$ ( $mll kappa $ ). For computing the matrix exponential, we obtain eigenvalues of the $mtimes m$ matrix using available library packages and compute an ordinary exponential function for the eigenvalues. The scheme involves mainly matrix-vector multiplications, and its convergence rate is generally $O(Delta t^{m-1})$ in time so that it allows taking a larger timestep size as $m$ increases. We demonstrate CPU time reduction compared with results from the five-stage fourth-order Runge–Kutta method for a certain accuracy. We also demonstrate error behaviors for long-time simulations. Case studies are also presented, showing loss of orthogonality that can be recovered by adding a low-cost reorthogonalization technique.