Stability and Accuracy of Semi-Extrapolated Finite Difference Schemes

When numerically solving partial differential equations, finite difference methods are a popular choice. Several factors come into play when choosing a finite difference method, such as stability, accuracy, and computational cost. In response to the small stability regions of explicit methods and the computational cost of implicit methods, we've developed a novel discretization technique called semi-extrapolation. Semi-extrapolation generates explicit schemes from implicit schemes by applying extrapolation in an unconventional fashion. Unlike extrapolation, which can severely curtail stability, semi-extrapolation can improve stability, as compared to analogous explicit methods. Furthermore, semi-extrapolation can have unexpected effects on accuracy. In this presentation, the concept of semi-extrapolation will be introduced and two semi-extrapolated discretizations of the Advection-Diffusion Equation will be discussed. Then, the accuracies and stabilities of these semi-extrapolated discretizations will be compared to the accuracies and stabilities of analogous mainstream finite difference discretizations.