On convergence of numerical solutions for the compressible MHD system with exactly divergence-free magnetic field

TL;DR

This paper develops a convergence theory for numerical solutions of the compressible MHD system that exactly preserve the divergence-free condition of the magnetic field, ensuring reliable simulations of magnetohydrodynamics.

Contribution

It introduces a generalized Lax equivalence framework for MHD schemes with divergence-free magnetic fields and proposes two novel mixed finite volume-finite element methods.

Findings

Proved dissipative weak-strong uniqueness principle for the system.

Established convergence of the proposed schemes to classical solutions.

Demonstrated the schemes' consistency and convergence through numerical analysis.

Abstract

We study a general convergence theory for the numerical solutions of compressible viscous and electrically conducting fluids with a focus on numerical schemes that preserve the divergence free property of magnetic field exactly. Our strategy utilizes the recent concepts of dissipative weak solutions and consistent approximations. First, we show the dissipative weak--strong uniqueness principle, meaning a dissipative weak solution coincides with a classical solution as long as they emanate from the same initial data. Next, we show the convergence of consistent approximation towards the dissipative weak solution and thus the classical solution. Upon interpreting the consistent approximation as the stability and consistency of suitable numerical solutions we have established a generalized Lax equivalence theory: convergence $\Longleftrightarrow$ stability and consistency. Further, to illustrate the application of this theory, we propose two novel mixed finite volume-finite element methods with exact divergence-free magnetic field. Finally, by showing solutions of these two schemes are consistent approximations, we conclude their convergence towards the dissipative weak solution and the classical solution.