An arbitrary high-order Spectral Difference method for the induction equation

Maria Han Veiga, David A. Velasco-Romero, Quentin Wenger, Romain Teyssier

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


We study in this paper three variants of the high-order Discontinuous Galerkin (DG) method with Runge-Kutta (RK) time integration for the induction equation, analysing their ability to preserve the divergence-free constraint of the magnetic field. To quantify divergence errors, we use a norm based on both a surface term, measuring global divergence errors, and a volume term, measuring local divergence errors. This leads us to design a new, arbitrary high-order numerical scheme for the induction equation in multiple space dimensions, based on a modification of the Spectral Difference (SD) method [1] with ADER time integration [2]. It appears as a natural extension of the Constrained Transport (CT) method. We show that it preserves ∇⋅B→=0 exactly by construction, both in a local and a global sense. We compare our new method to the 3 RKDG variants and show that the magnetic energy evolution and the solution maps of our new SD-ADER scheme are qualitatively similar to the RKDG variant with divergence cleaning, but without the need for an additional equation and an extra variable to control the divergence errors.

Original languageEnglish (US)
Article number110327
JournalJournal of Computational Physics
StatePublished - Aug 1 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


  • Divergence-free
  • High-order
  • Induction equation
  • Numerical analysis
  • Spectral Difference


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