Stability 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. Extrapolation can severely curtail stability, however, we've found that semi-extrapolation can improve stability, as compared to analogous explicit methods. In our presentation, we'll introduce our semi-extrapolation technique and discretize the Advection-Diffusion Equation according to semi-extrapolated methods and mainstream methods. Then, we'll discuss the stability regions of the schemes and analyze how the stabilities of the semi-extrapolated schemes compare to the stabilities of analogous schemes.