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

TL;DR

This paper introduces a new high-order Spectral Difference method for the induction equation that exactly preserves the divergence-free condition of magnetic fields, improving accuracy without additional divergence control equations.

Contribution

A novel high-order SD-ADER scheme that maintains divergence-free magnetic fields exactly, extending the Constrained Transport approach within a spectral difference framework.

Findings

The new SD-ADER scheme preserves divergence-free condition exactly.

It produces similar magnetic energy evolution as divergence cleaning RKDG methods.

The scheme does not require extra equations or variables for divergence control.

Abstract

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 $\nabla\cdot\vec{B}=0$ 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. [1] Liu Y., Vinokur M., Wang Z.J. (2006) Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids. In: Groth C., Zingg D.W. (eds) Computational Fluid Dynamics 2004. Springer, Berlin, Heidelberg [2] Dumbser M., Castro M., Par\'es C., Toro E.F (2009) ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows. In: Computers & Fluids, Volume 38, Issue 9