Numerical stability of the combined advection-diffusion equation with nonuniform spatial grid

The advection-diffusion equation is often solved by implicit, finite-difference schemes that are unconditionally stable when the grid interval is uniform. When such schemes are generalized to account for nonuniform grid spacing, instability can result. The cause of this difficulty is identified, and a procedure is given to reclaim stability. An example is provided to show that similar computational problems can be encountered in the use of explicit differencing schemes.