An efficient high-order time integration method for spectral-element discontinuous galerkin simulations in electromagnetics

Misun Min, Paul Fischer

Research output: Contribution to journalArticlepeer-review


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 κ × κ. 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 × m (m < κ). For computing the matrix exponential, we obtain eigenvalues of the m × 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 (Δ 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.

Original languageEnglish (US)
Pages (from-to)582-603
Number of pages22
JournalJournal of Scientific Computing
Issue number3
StatePublished - Dec 2013
Externally publishedYes


  • Arnoldi process
  • Exponential time integration
  • Krylov approximation
  • Matrix exponential
  • Spectral-element discontinuous Galerkin method

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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