Generalised summation-by-parts operators and variable coefficients

Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators

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.

Application of modal filtering to a spectral difference method

https://pure.mpg.de/pubman/item/item_3220027_1/component/file_3220028/arXiv%3A2001.00507.pdf

Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points

A recently introduced framework of semidiscretisations for hyperbolic conservation laws known as correction procedure via reconstruction (CPR, also known as flux reconstruction) is considered in the extended setting of summation-by-parts (SBP) operators using simultaneous approximation terms (SATs). This reformulation can yield stable semidiscretisations for linear advection and Burgers' equation as model problems. In order to enhance these properties, modal filters are introduced to this framework. As a second part of a series, the results of Ranocha, Glaubitz, Offner, and Sonar ("Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation", 2016) concerning artificial dissipation / spectral viscosity are extended, yielding fully discrete stable schemes. Additionally, a new adaptive strategy to compute the filter strength is introduced and different possible applications of modal filters are compared both theoretically and numerically.

Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: Modal filtering

We give a short overview of an extended Discontinuous Galerkin and Spectral Difference Method using PKD polynomials on triangular grids. A corresponding exponential filter is used to avoid oscillations near discontinuous solutions and to give some stabilization for nonsmooth testcases.

T. Sonar

Papers

An extended Discontinuous Galerkin and Spectral Difference Method with modal filtering