NDLI: Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations

Content Provider	EDP Sciences
Author	He Yang Fengyan Li
Abstract	In this paper, we consider an exactly divergence-free scheme to solve the magnetic induction equations. This problem is motivated by the numerical simulations of ideal magnetohydrodynamic (MHD) equations, a nonlinear hyperbolic system with a divergence-free condition on the magnetic field. Computational methods without satisfying such condition may lead to numerical instability. One class of methods, constrained transport schemes, is widely used as divergence-free treatments. So far there is not much analysis available for such schemes. In this work, we take an exactly divergence-free scheme proposed by [Li and Xu, J. Comput. Phys. 231 (2012) 2655–2675] as a candidate of the constrained transport schemes, and adapt it to solve the magnetic induction equations. For the resulting scheme applied to the equations with a constant velocity field, we carry out von Neumann analysis for numerical stability on uniform meshes. We also establish the stability and error estimates based on energy methods. In particular, we identify the stability mechanism due to the spatial and temporal discretizations, and the role of the exactly divergence-free property of the numerical solution for stability. The analysis based on energy methods can be extended to non-uniform meshes, and they can also be applied to the magnetic induction equations with a variable velocity field, which is more relevant to the MHD simulations.
Ending Page	993
Starting Page	965
Page Count	29
File Format	HTM / HTML PDF
ISSN	0764583X
Alternate Webpage(s)	https://www.esaim-m2an.org/articles/m2an/abs/2016/04/m2an150052/m2an150052.html
e-ISSN	12903841
Issue Number	4
Journal	ESAIM: Mathematical Modelling and Numerical Analysis
Volume Number	50
DOI	10.1051/m2an/2015061
Language	English
Publisher	EDP Sciences
Publisher Date	2016-07-01
Access Restriction	Subscribed
Rights Holder	© EDP Sciences, SMAI 2016
Subject Keyword	Stability error estimates ideal magnetohydrodynamic (MHD) equations constrained transport divergence-free discontinuous galerkin magnetic induction equations Stability / error estimates / ideal magnetohydrodynamic (MHD) equations / constrained transport / divergence-free / discontinuous galerkin / magnetic induction equations Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods Finite difference methods Error bounds Stability and convergence of numerical methods PDEs in connection with astronomy and astrophysics Initial-boundary value problems for first-order hyperbolic systems
Content Type	Text
Resource Type	Article

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