A comparison of numerical schemes to solve the magnetic induction eigenvalue problem in a spherical geometry

We compare various methods of solving the magnetic induction eigenvalue problem in a sphere, each using toroidal–poloidal decomposition and spherical harmonics, but with a different radial discretisation. In the case of quiescent flow where only diffusion acts upon the magnetic field, we benchmark numerical convergence against the analytic decay rates, and find that a Galerkin scheme based on Chebyshev polynomials with an associated projection chosen such that the diffusion operator is self-adjoint, exhibits the fastest convergence of the schemes described. The importance of the speed of convergence becomes heightened with the introduction of a non-quiescent flow because of the reduction in the magnetic field length scales. We find that sufficiently converged solutions are generally difficult to locate unless we use the optimal Galerkin scheme.