We present the whole basis of numerical method and useful formulae for general relativistic magnetohydrodynamic simulations in Kerr space-time.

This paper describes the numerical solution to the hyperbolic system of magnetohydrodynamic (MHD) equations. First, by pointing out the approximations involved, the deal MHD equations are presented. Next, the MHD waves as well as the associated shocks and discontinuities, are presented. Then, based on the hyperbolicity of the ideal MHD equations, the application of upwind schemes, which have been developed for hydrodynamics, is discussed to solve the equations numerically. As an definite example, one and multi-dimensional codes based on the Total Variation Diminishing scheme are presented. The treatment in the multi-dimensional code, which maintains ∇·B = 0, is described. Through tests, the robustness of the upwind schemes for MHDs is demonstrated.