A multidimensional code for isothermal magnetohydrodynamic flows in astrophysics

We present a multidimensional 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. The TVD scheme is a second-order-accurate extension of the Roe-type upwind scheme. Recipes for building the one-dimensional IMHD code, including the normalized right and left eigen-vectors of the IMHD Jacobian matrix, are presented. Then we have extended the one-dimensional code to a multidimensional IMHD code through a Strang-type dimensional splitting. In the multidimensional code, an explicit cleaning step has been included to eliminate nonzero ∇ · B at every time step. To test the code, IMHD shock tube problems, which encompass all the physical IMHD structures, have been constructed. One-dimensional and two-dimensional shock tube tests have shown that the code captures all the structures correctly without producing noticeable oscillations. Strong shocks are resolved sharply, but weaker shocks spread more. Numerical dissipation (viscosity and resistivity) has been estimated through the decay test of a two-dimensional Alfvén wave. It has been found to be slightly smaller than that of the adiabatic magnetohydrodynamic code based on the same scheme. As an example of astrophysical applications, we have simulated the nonlinear evolution of the two-dimensional Parker instability under a uniform gravity.