Application of Modal Filtering to a Spectral Difference Method

TL;DR

This paper introduces a spectral filtering approach for the spectral difference method using APK polynomials, providing new error bounds and stability analysis to improve accuracy and stability in solving hyperbolic conservation laws.

Contribution

It adapts spectral viscosity filtering to SD methods with APK polynomials, offering new error bounds and the first stability analysis for schemes with spectral filtering.

Findings

Spectral filtering stabilizes the SD scheme and reduces oscillations.

Choice of polynomial basis affects stability and accuracy.

Stability analysis identifies optimal polynomial and filter combinations.

Abstract

We adapt the spectral viscosity (SV) formulation implemented as a modal filter to a Spectral Difference Method (SD) solving hyperbolic conservation laws. In the SD Method we use selections of different orthogonal polynomials (APK polynomials). Furthermore we obtain new error bounds for filtered APK extensions of smooth functions. We demonstrate that the modal filter also depends on the chosen polynomial basis in the SD Method. Spectral filtering stabilizes the scheme and leaves weaker oscillations. Hence, the selection of the family of orthogonal polynomials on triangles and their specific modal filter possesses a positive influence on the stability and accuracy of the SD Method. In the second part, we initiate a stability analysis for a linear scalar test case with periodic initial condition to find the best selection of APK polynomials and their specific modal filter. To the best of our knowledge, this work is the first that gives a stability analysis for a scheme with spectral filtering. Finally, we demonstrate the influence of the underlying basis of APK polynomials in a well-known test case.