Application of Modal Filtering to a Spectral Difference Method

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.

Updated version

References in corpus (3)

• Orthogonal polynomials of several variables
• Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation
• Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: Modal filtering

Cited by in corpus (13)

• Positivity-Preserving Entropy-Based Adaptive Filtering for Discontinuous Spectral Element Methods
• Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
• Smooth and compactly supported viscous sub-cell shock capturing for Discontinuous Galerkin methods
• DeC and ADER: Similarities, Differences and a Unified Framework
• OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws
• Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points
• Shock Capturing by Bernstein Polynomials for Scalar Conservation Laws
• Towards Stable Radial Basis Function Methods for Linear Advection Problems
• High Order Edge Sensors with ${\ell }^1$ Regularization for Enhanced Discontinuous Galerkin Methods
• Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions
• Error boundedness of Correction Procedure via Reconstruction / Flux Reconstruction
• Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: Modal filtering
• A Riemann Difference Scheme for Shock Capturing in Discontinuous Finite Element Methods