Numerical Methods for the Magnetic Induction Equation with Hall Effect and Projections onto Divergence-Free Vector Fields

TL;DR

This paper develops and analyzes high-order stable numerical schemes for the nonlinear magnetic induction equation with Hall effect, including novel boundary conditions and divergence correction methods, ensuring reliable simulations in plasma physics.

Contribution

It introduces energy-stable schemes with new boundary conditions and divergence correction techniques for the Hall induction equation, advancing numerical modeling accuracy.

Findings

Energy stability of proposed schemes demonstrated

New outflow boundary conditions improve energy estimates

Projection methods effectively correct divergence errors

Abstract

The nonlinear magnetic induction equation with Hall effect can be used to model magnetic fields, e.g. in astrophysical plasma environments. In order to give reliable results, numerical simulations should be carried out using effective and efficient schemes. Thus, high-order stable schemes are investigated here. Following the approach provided recently by Nordstr\"om (J Sci Comput 71.1, pp. 365--385, 2017), an energy analysis for both the linear and the nonlinear induction equation including boundary conditions is performed at first. Novel outflow boundary conditions for the Hall induction equation are proposed, resulting in an energy estimate. Based on an energy analysis of the initial boundary value problem at the continuous level, semidiscretisations using summation by parts (SBP) operators and simultaneous approximation terms are created. Mimicking estimates at the continuous level, several energy stable schemes are obtained in this way and compared in numerical experiments. Moreover, stabilisation techniques correcting errors in the numerical divergence of the magnetic field via projection methods are studied from an energetic point of view in the SBP framework. In particular, the treatment of boundaries is investigated and a new approach with some improved properties is proposed.