The first part of this paper reviews some issues representing major computational challenges for global MHD models of the space environment. These issues include mathematical formulation and discretization of the governing equations that ensure the proper jump conditions and propagation speeds, regions of relativistic Alfve´n speed, and controlling the divergence of the magnetic field. The second part of the paper concentrates on modern solution methods that have been developed by the aerodynamics, applied mathematics and DoE communities. Such methods have recently begun to be implemented in space‐physics codes, which solve the governing equations for a compressible magnetized plasma. These techniques include high‐resolution upwind schemes, block‐based solution‐adaptive grids and domain decomposition for parallelization. We describe the space physics MHD code developed at the University of Michigan, based on the developments listed above. © 2003 American Institute of Physics