Application of Modal Filtering to a Spectral Difference Method
This work provides the first stability analysis for a spectral filtering scheme, offering guidance for selecting polynomial bases in SD methods to improve stability and accuracy.
The authors adapt spectral viscosity (SV) modal filtering to the Spectral Difference Method (SD) for hyperbolic conservation laws, using APK orthogonal polynomials. They obtain new error bounds for filtered APK extensions and show that the modal filter stabilizes the scheme while reducing oscillations, with the choice of polynomial basis affecting stability and accuracy.
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.