Summation-by-parts operators for general function spaces: optimal nodes

TL;DR

This paper extends the concept of optimal summation-by-parts operators from polynomial spaces to general function spaces using generalized Gauss-Lobatto quadrature, providing algorithms and demonstrating their effectiveness.

Contribution

It introduces a method to find optimal SBP nodes for general function spaces, broadening the applicability beyond polynomial-based approaches.

Findings

Generalized Gauss-Lobatto quadrature yields optimal SBP nodes for various function spaces.

The proposed algorithm computes accurate and efficient quadrature rules.

Application to boundary value problems demonstrates practical utility.

Abstract

Gauss-Lobatto quadrature nodes and weights are optimal for closed summation-by-parts (SBP) formulations based on polynomial approximation spaces in the sense that for a prescribed function space they yield an SBP operator of minimal dimension. We show that the same principle extends to general (possibly non-polynomial) function spaces: an associated generalised Gauss-Lobatto quadrature provides the optimal nodes and weights for the SBP formulation. We present an algorithm for computing these quadrature rules, demonstrate their accuracy and efficiency across a range of function spaces, and illustrate their use in solving initial boundary value problems.