Stable filtering procedures for nodal discontinuous Galerkin methods

TL;DR

This paper proves the stability of a common filtering procedure in nodal discontinuous Galerkin methods, linking it to polynomial basis functions and high-order quadrature, and validates the results through numerical tests.

Contribution

It establishes the stability of filtering in DG methods by connecting it to a contractivity condition and provides a theoretical framework for understanding filtering stability.

Findings

Filtering is stable under the proven conditions.

Stability is linked to a contractivity condition.

Numerical tests confirm theoretical results.

Abstract

We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre-Gauss-Lobatto quadrature. The theoretical discussion serves to re-contextualize stable filtering results for finite difference methods into the DG setting. It is shown that the stability of the filtering is equivalent to a particular contractivity condition borrowed from the analysis of so-called transmission problems. As such, the temporal stability proof relies on the fact that the underlying spatial discretization of the problem possesses a semi-discrete bound on the solution. Numerical tests are provided to verify and validate the underlying theoretical results.