OPTIMAL EXPONENTIAL DIFFERENCE SCHEME FOR SOLVING PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract

The existing single-step finite-difference schemes for solving time-dependent parabolic partial differential equations become, due to the inadequate approximation of the time derivative, either inaccurate or unstable, or introduce spurious oscillations in the solution when the time step is relatively large. An optimal single-step exponential scheme is developed herein based on a study of the analytic behavior of time variation of the time derivative. Its excellent performance is demonstrated by its application to the one-dimensional transient diffusion equation. The comparison among a class of commonly used existing single-step schemes shows that it is by far the most accurate for a wide range of time steps. Its performance, the author believes, could be even more optimized if it is utilized in the context of an algorithm that automatically varies the time step based on the local truncation error. Generalizations to multidimensional problems are also discussed.