Robust Preconditioners for Incompressible MHD Models

TL;DR

This paper introduces two robust preconditioners for discretized incompressible MHD models, improving solver efficiency and preserving physical constraints, with proven theoretical robustness and supporting numerical experiments.

Contribution

It develops new block preconditioners for MHD systems that are robust across parameters and applicable to various discretizations, with improved theoretical analysis.

Findings

Preconditioners are robust against physical and discretization parameters.

Krylov methods with these preconditioners preserve divergence-free conditions.

Numerical results confirm theoretical robustness.

Abstract

In this paper, we develop two classes of robust preconditioners for the structure-preserving discretization of the incompressible magnetohydrodynamics (MHD) system. By studying the well-posedness of the discrete system, we design block preconditioners for them and carry out rigorous analysis on their performance. We prove that such preconditioners are robust with respect to most physical and discretization parameters. In our proof, we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters, which are usually difficult to choose in practice. This new technique is not only applicable to the MHD system, but also to other problems. Moreover, we prove that Krylov iterative methods with our preconditioners preserve the divergence-free condition exactly, which complements the structure-preserving discretization. Another feature is that we can directly generalize this technique to other discretizations of the MHD system. We also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners.