Abstract
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 \(m\times m\) (\(m\ll \kappa \)). For computing the matrix exponential, we obtain eigenvalues of the \(m\times m\) 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.
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This work was supported by the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract DE-AC02-06CH11357.
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Min, M., Fischer, P. An Efficient High-Order Time Integration Method for Spectral-Element Discontinuous Galerkin Simulations in Electromagnetics. J Sci Comput 57, 582–603 (2013). https://doi.org/10.1007/s10915-013-9718-8
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DOI: https://doi.org/10.1007/s10915-013-9718-8