## 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 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 language | English (US) |
---|---|

Pages (from-to) | 582-603 |

Number of pages | 22 |

Journal | Journal of Scientific Computing |

Volume | 57 |

Issue number | 3 |

DOIs | |

State | Published - Dec 2013 |

Externally published | Yes |

## Keywords

- 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