A high-order iterative implicit-explicit hybrid scheme for magnetohydrodynamics

An iterative implicit-explicit hybrid scheme is proposed for one-dimensional ideal magnetohydrodynamical (MHD) equations. The scheme is accurate to second order in both space and time for all Courant numbers. Each kind of MHD wave, both compressible and incompressible, may be either implicitly or explicitly, or partially implicitly and partially explicitly treated depending on their associated Courant numbers in each numerical cell, and the scheme is able to smoothly switch between explicit and implicit calculations. The scheme is of Godunov type in both explicit and implicit regimes, in which the flux needed in a Godunov scheme is calculated from Riemann problems, is in a strictly conservation form, and is able to resolve MHD discontinuities. Only a single level of iterations is involved in the scheme, and the iterations solve both the implicit relations arising from upstream centered differences for all wave families and the nonlinearity of MHD equations. Multicolors proposed in the paper may significantly reduce the number of iterations required to reach a converged solution. Compared with the largest Courant number in a simulation, only a small number of iterations are needed in the scheme. The feature of the scheme has been shown through numerical examples. It is easy to vectorize the computer code of the scheme.