Eigenvalue enclosures and convergence for the linearized MHD operator

TL;DR

This paper presents methods for computing certified eigenvalue enclosures of linear magnetohydrodynamics operators in different configurations, achieving high accuracy and establishing convergence rates for the approaches.

Contribution

It introduces new techniques for eigenvalue enclosures in MHD operators, including a Schur complement formulation and a specialized Zimmermann-Mertins method, with proven convergence.

Findings

High-accuracy eigenvalue bounds for plane slab configurations.

Applicability of the Zimmermann-Mertins technique to cylindrical configurations.

Established convergence rates for both methods.

Abstract

We discuss how to compute certified enclosures for the eigenvalues of benchmark linear magnetohydrodynamics operators in the plane slab and cylindrical pinch configurations. For the plane slab, our method relies upon the formulation of an eigenvalue problem associated to the Schur complement, leading to highly accurate upper bounds for the eigenvalue. For the cylindrical configuration, a direct application of this formulation is possible, however, it cannot be rigourously justified. Therefore in this case we rely on a specialized technique based on a method proposed by Zimmermann and Mertins. In turns this technique is also applicable for finding accurate complementary bounds in the case of the plane slab. We establish convergence rates for both approaches.