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The Type Ia supernova progenitor problem is one of the most perplexing andexciting problems in astrophysics, requiring detailed numerical modeling tocomplement observations of these explosions. One possible progenitor that hasmerited recent theoretical attention is the white dwarf merger scenario, whichhas the potential to naturally explain many of the observed characteristics ofType Ia supernovae. To date there have been relatively few self-consistentsimulations of merging white dwarf systems using mesh-based hydrodynamics. Thisis the first paper in a series describing simulations of these systems using ahydrodynamics code with adaptive mesh refinement. In this paper we describe ournumerical methodology and discuss our implementation in the compressiblehydrodynamics code CASTRO, which solves the Euler equations, and the Poissonequation for self-gravity, and couples the gravitational and rotation forces tothe hydrodynamics. Standard techniques for coupling gravitation and rotationforces to the hydrodynamics do not adequately conserve the total energy of thesystem for our problem, but recent advances in the literature allow progressand we discuss our implementation here. We present a set of test problemsdemonstrating the extent to which our software sufficiently models a systemwhere large amounts of mass are advected on the computational domain over longtimescales. Future papers in this series will describe our treatment of theinitial conditions of these systems and will examine the early phases of themerger to determine its viability for triggering a thermonuclear detonation.

White Dwarf Mergers on Adaptive Meshes I. Methodology and Code Verification