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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:438

Article

Veiga, Maria Han¹  Velasco-Romero, David A.²  Wenger, Quentin³  Teyssier, Romain² 

[1] Univ Michigan, Michigan Inst Data Sci, Weiser Hall 500 Church St,Suite 600, Ann Arbor, MI 48109 USA

[2] Univ Zurich, Inst Computat Sci, Winterthurerstr 190, CH-8057 Zurich, Switzerland

[3] Swiss Fed Inst Technol, Wolfgang Pauli Str 27, CH-8093 Zurich, Switzerland

关键词: Numerical analysis;    High-order;    Divergence-free;    Induction equation;    Spectral Difference;   

DOI  :  10.1016/j.jcp.2021.110327

来源: Elsevier

【 摘 要 】

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. del . (B) over right arrow = 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. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free