Implicit kinetic schemes for the ideal MHD equations and their parallel implementation

Abstract

The implicit kinetic schemes, namely the kinetic flux-vector-split (KFVS) and the kinetic wave/particle split (KWPS) schemes are derived for the Euler and ideal MHD equations. However, to date, all the papers on kinetic schemes reported in the literature employ an explicit formulation because of its simplicity. The major drawback of the explicit formulation is the restriction placed on the time step allowed for the stability of the scheme. This new splitting does not require the evaluation of error functions, increasing computational speed. They showed that for steady state calculations, the implicit kinetic schemes were significantly more efficient than the explicit schemes while retaining the accuracy and robustness of explicit schemes. Equations on numerical tests show that the implicit schemes retain the accuracy and robustness of the explicit schemes while increasing their efficiency by reducing the number of time steps required in obtaining the numerical solution. Excellent results are obtained for 2D cylindrical blast wave and flow past a blunt body with and without the magnetic field. The projection scheme has been found to be an expensive approach numerically, especially in generalized coordinate system.