NUMERICAL MAGNETOHYDRODYNAMICS IN ASTROPHYSICS - ALGORITHM AND TESTS FOR ONE-DIMENSIONAL FLOW

We describe a numerical code to solve the equations for ideal magnetoh ydrodynamics (MHD). It is based on an explicit finite difference schem e on an Eulerian grid, called the total variation diminishing (TVD) sc heme, which is a second-order-accurate extension of the Roe-type upwin d scheme. We also describe a nonlinear Riemann solver for ideal MHD, w hich includes rarefactions as well as shocks. The numerical code and t he Riemann solver have been used to test each other. Extensive tests e ncompassing all the possible ideal MHD structures with planar symmetri es (i.e., one-dimensional flows) are presented. These include those fo r which the field structure is two dimensional (i.e., those flows ofte n called ''1 + 1/2 dimensional'') as well as those for which the magne tic field plane rotates (i.e., those flows often called ''1 + 1/2 + 1/ 2 dimensional''). Results indicate that the code can resolve strong fa st, slow, and magnetosonic shocks within two to four cells, but more c ells are required if shocks become weak. With proper steepening, we co uld resolve rotational discontinuities within three to five cells. How ever, without successful implementation of steepening, contact discont inuities are resolved with similar to 10 cells and tangential disconti nuities are resolved with similar to 15 cells. Our tests confirm that slow compound structures with two-dimensional magnetic fields are comp osed of intermediate shocks (so-called 2-4 intermediate shocks) follow ed by slow rarefaction waves. Finally, tests demonstrate that in two-d imensional magnetohydrodynamics, fast compound structures, which are c omposed of intermediate shocks (so-called 1-3 intermediate shocks) pre ceded by fast rarefaction waves, are also possible.