  The Type Ia supernova progenitor problem is one of the most perplexing and
exciting problems in astrophysics, requiring detailed numerical modeling to
complement observations of these explosions. One possible progenitor that has
merited recent theoretical attention is the white dwarf merger scenario, which
has the potential to naturally explain many of the observed characteristics of
Type Ia supernovae. To date there have been relatively few self-consistent
simulations of merging white dwarf systems using mesh-based hydrodynamics. This
is the first paper in a series describing simulations of these systems using a
hydrodynamics code with adaptive mesh refinement. In this paper we describe our
numerical methodology and discuss our implementation in the compressible
hydrodynamics code CASTRO, which solves the Euler equations, and the Poisson
equation for self-gravity, and couples the gravitational and rotation forces to
the hydrodynamics. Standard techniques for coupling gravitation and rotation
forces to the hydrodynamics do not adequately conserve the total energy of the
system for our problem, but recent advances in the literature allow progress
and we discuss our implementation here. We present a set of test problems
demonstrating the extent to which our software sufficiently models a system
where large amounts of mass are advected on the computational domain over long
timescales. Future papers in this series will describe our treatment of the
initial conditions of these systems and will examine the early phases of the
merger to determine its viability for triggering a thermonuclear detonation.
