Numerical magetohydrodynamics in astronphysics: Algorithm and tests for one-dimensional flow`

We describe a numerical code to solve the equations for ideal magnetohydrodynamics (MHD). It is based on an explicit finite difference scheme on an Eulerian grid, called the total variation diminishing (TVD) scheme, which is a second-order-accurate extension of the Roe-type upwind scheme. We also describe a non-linear Riemann solver for ideal MHD, which includes rarefractions as well as shocks. The numerical code and the Riemann solver have been used to test each other. Extensive tests encompassing all the possible ideal MHD structures with planar symmetries (i.e., one-dimensional flows) are presented. These include those for which the field structure is two dimensional (i.e., those flows often called '1 + 1/2 dimensional') as well as those for which the magnetic field plane rotates (i.e., those flows often called '1 + 1/2 + 1/2 dimensional'). Results indicate that the code can resolve strong fast, slow, and magnetosonic shocks within two to four cells, but more cells are required if shocks become weak. With proper steepening, we could resolve rotational discontinuities within three to five cells. However, without successful implementation of steepening, contact discontinuities are resolved with approximately 10 cells and tangential discountinuities are resolved with approximately 15 cells. Out tests confirm that slow compound structures with tow-dimensional magnetic fields are composed of intermediate shocks (so-called 2-4 intermediate shocks) followed by slow rarefaction waves. Finally, tests demostrate that in two-dimensional magnetohydrodynamics, fast compound structures, which are composed of intermediate shocks (so-called 1-3 intermediate shocks) preceeded by fast rarefaction waves, are also possible.