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

• Published in: Journal of computational physics 2021-08, Vol.438, p.110327, Article 110327
• Main Authors: Han Veiga, Maria, Velasco-Romero, David A., Wenger, Quentin, Teyssier, Romain
• Format: Article
• Language: English
• Subjects: Computational physics; Constraints; Divergence; Divergence-free; High-order; Induction equation; Numerical analysis; Runge-Kutta method; Spectral Difference; Time integration
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110327

•SD-ADER is a new arbitrary high-order method for the induction equation.•It can be seen as a high-order extension of the Constrained Transport (CT) method.•In SD-ADER, the B-field is provably divergence-free (both locally and globally).•Numerical results of SD-ADER are similar to the RKDG with divergence cleaning.•SD-ADER does not need additional equations/variables to control the divergence. 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.