Spectral Basis Functions for Ideal and Extended MHD

The ability to reliably reproduce and distinguish various modes of ideal and extended MHD in numerical computation depends sensitively on the choice of spatial representation. The original development of ideal-MHD eigenvalue solvers [Gruber and Rappaz, Springer-Verlag (1985), for example] provides a numerical foundation. However, the interactions of transport with magnetic topology evolution in nonlinear simulations impose additional criteria that favor high-order and spectral representations. We compare global spectral representations and spectral-element representations for several possible systems of variables in ideal and non-ideal cylindrical eigenvalue computations. While many of the conclusions from the original low-order eigenvalue computations hold for second-order systems for displacement, first-order systems for all physical components are more representative of time- dependent computations and have distinct numerical properties. As expected, global representations converge slowly for localized modes, but placing borders of spectral elements at rational surfaces leads to rapid convergence.