Eigenvalue enclosures and convergence for the linearized MHD operator
Provides rigorous eigenvalue bounds for benchmark MHD stability problems, enabling reliable validation of numerical methods.
The paper presents certified eigenvalue enclosures for linearized MHD operators in plane slab and cylindrical pinch configurations, achieving highly accurate upper bounds with proven convergence rates.
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.