  A generalization of implicit conservative numerics to multiple dimensions
requires advanced concepts of tensor analysis and differential geometry and
hence a more thorough dedication to mathematical fundamentals than maybe
expected at first glance. Hence we begin to discuss fundamental mathematics and
physics of RHD with special focus on differential geometric consistency and
study numerical methods for nonlinear conservation laws to gain a solid
definition of the term conservative. The efforts in tensor analysis will be
needed when applying Vinokurs theorem to gain the strong conservation form for
conservation laws in general curvilinear coordinates. Moreover, it will be
required to slightly reformulate the artificial viscosity for such nonlinear
coordinates. Astronomical objects are characterized by fast flows and high
propagation speeds on the one hand but astronomical length and time scales on
the other hand. Implicit numerical schemes are not affected by the Courant
Friedrichs Levy condition which limits explicit schemes to rather impracticably
small time steps. Implicit methods however produce algebraic problems that
require matrix inversion which is computationally expensive. In order to
achieve viable resolution, adaptive grid techniques have been developed. It is
desired to treat processes on small length scales like shocks and ionization
fronts as well as physics at the extent of the objects dimension itself like
large scale convection flows and pulsations. The combination of implicit
schemes and adaptive grids allows to resolve astrophysics appropriately at
various scales. In the last chapter of this paper we study problem oriented
adaptive grid generation in 2D and 3D. We establish three main postulations for
an ideal grid and analyze several feasible approaches.
