Semi-Extrapolated Finite Difference Schemes: Accuracy and Consistency

When 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, computational cost, accuracy, and consistency. In response to the small stability regions of explicit methods and the computational cost of implicit methods, we’ve developed a novel discretization technique (semi-extrapolation) that generates explicit schemes from implicit schemes by applying extrapolation to the implicit schemes in an unconventional fashion. Semi-extrapolating can lead to improved stabilities as compared to the stabilities of analogous explicit schemes, however, consistency and accuracy can be affected by semi-extrapolation. In our presentation, we’ll discuss our semi-extrapolation technique and introduce several semi-extrapolated discretizations of the Advection Equation and the Advection-Diffusion Equation. We’ll then analyze the consistency of these semi-extrapolated discretizations, compare their accuracies against the accuracies of several common discretizations, and discuss how stability constraints and choice of extrapolation stencil both influence the consistency and accuracy of semi-extrapolated schemes.