Summation-by-parts operators for general function spaces: The second derivative

TL;DR

This paper extends the concept of function-space summation-by-parts (FSBP) operators to second derivatives, enabling stable, high-order numerical solutions for PDEs using diverse function spaces beyond polynomials.

Contribution

It introduces a new class of second-derivative FSBP operators that are applicable to various function spaces, broadening the scope of SBP methods for PDEs.

Findings

Operators maintain mimetic properties of polynomial SBP operators.

Constructed operators for trigonometric, exponential, and radial basis functions.

Framework ensures existence and straightforward construction of second-derivative FSBP operators.

Abstract

Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.