Adaptive Mesh Refinement, or AMR, has been used as a tool in CFD to better resolve and capture compressible flows, both for continuum solvers [5, 4, 8] and DSMC (Direct Simulation Monte Carlo) solvers [6, 7, 21]. Presented is the True Direction Equilibrium Flux Method, or TDEFM [1, 2, 3] applied with an adaptive meshing technique and diffusely reflecting boundary conditions. Continuum methods generally transfer fluxes between directly adjacent cells only; TDEFM is capable of transferring fluxes of mass, momentum and energy from any source cell to any given destination cell. TDEFM has been previously shown to provide superior results when compared to existing continuum solvers [1, 2, 3] for unaligned flows on cartesian grids. In the present method, cells are isotropically divided in order to more accurately resolve the flow field. Various mesh adaption parameters are employed, taken from both continuum and direct solver applications on adaptive meshes. Results have shown that use of an adaptive mesh with an adaptation parameter based on the local mean free path l has reduced computational requirements considerably while maintaining resolution of important features of the flow.