Extension of the piecewise parabolic method to multidimensional ideal magnetohydrodynamics

An extension of the piecewise parabolic method to treat multidimensional ideal magnetohydrodynamical equations is presented in this paper. The multidimensional scheme is constructed from a one-dimensional functioning code based on the dimensional splitting method originally suggested by Strang. The functioning code is built upon a nonlinear Riemann solver for ideal MHD equations recently developed by the authors. The correctness of the scheme is tested in the steepening of waves in both one- and two-dimensional situations and in various MHD shock-tube problems which involve all the discontinuities in ideal MHD. The robust character of the scheme is demonstrated in the shock-tube problems and in the interaction between MHD shocks and a cloud. The results of these problems show that the scheme keeps the principal advantages of a high-order Godunov scheme: Robust operation in the presence of very strong waves, thin shock fronts with little attendant noise generation, and thin contact discontinuity.