Profiles spanning the Lagrangian-homologous grid transition region in a ﹩0.6{M}_{\odot }﹩ WD with a core temperature of 10⁷ K undergoing accretion of solar composition material at a rate of ﹩{10}^{-10}{M}_{\odot }﹩ yr⁻¹. At the time shown ﹩{M}_{\mathrm{acc}}/{M}_{\mathrm{ign}}=0.2﹩. Shown for comparison are the results of the treatment discussed here for a 5000 years timestep and a 1000 years timestep and the treatment discussed in Paper II. For the current treatment the transition from homologous to Lagrangian is indicated by two open circles at either end of the region. The location ﹩\delta M﹩ in from the surface, the base of material newly accreted this step, is indicated by the triangle.