Computing SIAC spline coefficients

TL;DR

This paper derives simple formulas for computing optimal SIAC spline coefficients used to enhance the accuracy of Discontinuous Galerkin solutions through convolution with B-splines, achieving superconvergence.

Contribution

It introduces straightforward formulas for calculating SIAC spline coefficients that optimize polynomial reproduction in DG post-processing.

Findings

Derived formulas for optimal SIAC spline coefficients

Enhanced superconvergence in DG methods

Improved accuracy of polynomial reproduction

Abstract

The Discontinuous Galerkin (DG) method applied to hyperbolic differential equations outputs weakly-linked polynomial pieces. Post-processing these pieces by Smoothness-Increasing Accuracy-Conserving (SIAC) convolution with B-splines can improve the accuracy of the output and yield superconvergence. SIAC convolution is considered optimal if the SIAC kernels, in the form of a linear combinations of B-splines of degree d, reproduce polynomials of degree 2d. This paper derives simple formulas for computing the optimal SIAC spline coefficients.