On the Stability of Boundary Conditions for Separable Difference Approximations to Parabolic Equations

The influence of discrete boundary conditions on the stability of a finite difference scheme is difficult to analyze completely. Extraneous eigenvalues may be introduced, and their location is difficult to trace. In this paper, we consider approximating parabolic equations by finite-difference schemes with a single one-level spatial operator, and arbitrary time-differencing. (We call such schemes separable.) For discrete boundary conditions based on extrapolation (of arbitrary order), we show how to explicitly characterize these extraneous eigenvalues and thus guarantee stability, Using this, we completely verify the stability of compact fourth and sixth order schemes, and give numerical results indicating the best order of extrapolation to use.