On a semi-implicit scheme for spectral simulations of dispersive magnetohydrodynamics

The effectiveness of a semi-implicit (SI) temporal scheme is discussed in the context of the dispersive magnetohydrodynamics where, due to the whistler modes, stability of explicit algorithms requires a time step decreasing quadratically as the resolution is linearly increased. After analyzing the effects of this scheme on the Alfvén-wave dispersion relation, spectral simulations of nonlinear initial value problems where small-scale dispersion has a main effect on the global dynamics are presented. Permitting a moderate, albeit significant, increase of the time step for a minor additional cost relatively to explicit schemes, the SI algorithm provides an efficient tool in situations, such as turbulent regimes, where the time steps making fully implicit schemes efficient are too large to ensure a satisfactory accuracy.