Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics

Bernardo Cockburn, Shukai Du, Manuel A. Sánchez

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces. These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy. We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy. Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time. The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system. The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time. The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables.

Original languageEnglish (US)
Pages (from-to)775-796
Number of pages22
JournalComputational Methods in Applied Mathematics
Volume22
Issue number4
DOIs
StatePublished - Oct 1 2022

Bibliographical note

Publisher Copyright:
© 2022 Walter de Gruyter GmbH, Berlin/Boston.

Keywords

  • Discontinuous Galerkin Methods
  • Energy-Conserving Methods
  • Time-Dependent Maxwell's Equations

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