Explicit Least-degree Boundary Filters for Discontinuous Galerkin

TL;DR

This paper introduces explicit least-degree PSIAC boundary filters for Discontinuous Galerkin methods, which outperform existing filters in accuracy, stability, and efficiency at domain boundaries for hyperbolic equations.

Contribution

It presents new minimal-support, least-degree PSIAC filters with explicit symbolic forms that improve boundary accuracy and computational stability in DG methods.

Findings

Least-degree PSIAC filters outperform existing boundary filters.

Boundary error is reduced below interior symmetric filters.

Filters are stable, efficient, and easy to differentiate.

Abstract

Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the top-of-the-line one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, convolution with PSIAC filters reduces to a stable inner product with the DG output and hence provides a stable and efficient implementation. The paper focuses on the remarkable fact that, for the canonical linear or non-linear hyperbolic test equation, new piecewise constant PSIAC filters of small support outperform the top-of-the-line boundary filters in the literature. These least-degree filters reduce the error at the boundaries to less than the error even of the symmetric filters of the same support that are applied in the interior . Due to their simplicity, and since this least degree filter has an exact symbolic form, convolution is stable as well as efficient and derivatives of the convolved output are easy to compute.