A Multi-dimensional Code for Isothermal Magnetohydrodynamic Flows in Astrophysics

We present a multi-dimensional numerical code to solve isothermal magnetohydrodynamic (IMHD) equations for use in modeling astrophysical flows. First, we have built a one-dimensional code which is based on an explicit finite-difference method on an Eulerian grid, called the total variation diminishing (TVD) scheme. Recipes for building the one-dimensional IMHD code, including the normalized right and left eigenvectors of the IMHD Jacobian matrix, are presented. Then, we have extended the one-dimensional code to a multi-dimensional IMHD code through a Strang-type dimensional splitting. In the multi-dimensional code, an explicit cleaning step has been included to eliminate non-zero $\nabla\cdot B$ at every time step. To estimate the proformance of the code, one- and two-dimensional IMHD shock tube tests, and the decay test of a two-dimensional Alfv\'{e}n wave have been done. As an example of astrophysical applications, we have simulated the nonlinear evolution of the two-dimensional Parker instability under a uniform gravity.