Many problems in the physical sciences can be reduced to the solution of a system of time-dependent partial differential equations. Of particular interest to us are problems in combustion and heat and mass transfer, i.e., hydrocarbon ignition and catalytic combustion. The governing equations in these applications can be formulated as a system of parabolic mixed initial-boundary value problems. The numerical stiffness that results from solving these problems on a discrete mesh combined with the inherent stiffness of the disparate decay rates of the various chemical species necessitate the use of implicit time differencing methods. The problems also produce solutions that contain regions of high spatial activity, i.e., sharp peaks and steep fronts. Although an equispaced mesh could be used in the calculations, it is often more efficient to employ a nonuniform adaptive grid in the anticipation that the high activity regions will be better resolved. Important questions in such studies are the effects that adaptive time and space steps (both fixed and variable numbers of points) have on the stability and convergence of the parabolic solver. We investigate these issues in this paper for a class of linear mixed initial-boundary value problems.