Non-uniform Discontinuous Galerkin Filters via Shift and Scale

TL;DR

This paper introduces a unified framework for position-dependent spline filters in Discontinuous Galerkin methods, enabling efficient, stable boundary filtering through rational knot sequences and symbolic expressions, with demonstrated applications.

Contribution

It develops a general class of PSIAC filters with rational knot sequences, providing symbolic formulas and efficient convolution methods for boundary filtering in DG computations.

Findings

Rational knot sequences lead to rational filter coefficients.

Single dot product convolution improves stability and efficiency.

Framework applies to multiple established and new boundary filters.

Abstract

Convolving the output of Discontinuous Galerkin (DG) computations with symmetric Smoothness-Increasing Accuracy-Conserving (SIAC) filters can improve both smoothness and accuracy. To extend convolution to the boundaries, several one-sided spline filters have recently been developed. We interpret these filters as instances of a general class of position-dependent spline filters that we abbreviate as PSIAC filters. These filters may have a non-uniform knot sequence and may leave out some B-splines of the sequence. For general position-dependent filters, we prove that rational knot sequences result in rational filter coefficients. We derive symbolic expressions for prototype knot sequences, typically integer sequences that may include repeated entries and corresponding B-splines, some of which may be skipped. Filters for shifted or scaled knot sequences are easily derived from these prototype filters so that a single filter can be re-used in different locations and at different scales. Moreover, the convolution itself reduces to executing a single dot product making it more stable and efficient than the existing approaches based on numerical integration. The construction is demonstrated for several established and one new boundary filter.