M. Min and P. Fischer, "An Efficient High-Order Time Integration Method for Spectral-Element Discontinuous Galerkin Simulations in Electromagnetics," Preprint ANL/MCS-P1830-0111, January 2011. [pdf]
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 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 nxn. We project the matrix exponential into a small Krylov subspace by the Arnoldi process based on the modified Gram-Schumidt algorithm and perform a matrix exponential operation with a much smaller matrix of dimension m x m (m ≪ n). For computing the matrix exponential, we obtain eigenvalues of the m x 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[sup 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.