Summation-by-parts operators for general function spaces

TL;DR

This paper extends the theory of summation-by-parts (SBP) operators to general function spaces beyond polynomials, enabling their application to a broader range of approximation methods for differential equations.

Contribution

It develops a unified theory for SBP operators based on general function spaces, broadening their applicability beyond polynomial approximations.

Findings

Most results for polynomial SBP operators carry over to general function spaces.

SBP operators can be constructed using trigonometric, exponential, and radial basis functions.

The approach enhances the flexibility of stable, high-order numerical methods.

Abstract

Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.