An augmented Lagrangian preconditioner for the magnetohydrodynamics equations at high Reynolds and coupling numbers

TL;DR

This paper introduces a scalable augmented Lagrangian preconditioner for the MHD equations that remains robust at high Reynolds and coupling numbers, enabling efficient solutions for complex, coupled electromagnetic and hydrodynamic problems.

Contribution

The authors develop a novel, parameter-robust preconditioning method for the MHD equations that ensures divergence-free solutions and performs well in high Reynolds and coupling regimes.

Findings

Achieves robust performance for high Reynolds and coupling numbers in 2D and 3D

Extends to fully implicit time-dependent problems with stable solutions

Numerical experiments confirm robustness up to Reynolds numbers of 10,000 and 100,000

Abstract

The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretization of the $\mathbf{B}$-$\mathbf{E}$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. We extend our method to fully implicit methods for time-dependent problems which we solve robustly in both two and three dimensions. Our approach relies on specialized parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. We confirm the robustness of our solver by numerical experiments in which we consider fluid and magnetic Reynolds numbers and coupling numbers up to 10,000 for stationary problems and up to 100,000 for transient problems in two and three dimensions.