Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

TL;DR

This paper extends the theory of summation-by-parts (SBP) operators to multi-dimensional spaces based on general function spaces, broadening their applicability for stable, high-order numerical methods in differential equations.

Contribution

It develops a comprehensive theory for multi-dimensional function-space SBP operators, including their existence, construction, and properties, generalizing polynomial-based SBP methods.

Findings

Most results for polynomial SBP operators extend to general function spaces.

Multi-dimensional FSBP operators can improve accuracy and stability of numerical methods.

The theory connects SBP operators with quadrature and mimetic properties.

Abstract

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.