An optimization-based construction procedure for function space based summation-by-parts operators on arbitrary grids

TL;DR

This paper presents an optimization-based method for constructing function space summation-by-parts operators on arbitrary grids, improving stability and accuracy over traditional methods.

Contribution

It introduces a novel simultaneous optimization approach for constructing FSBP operators, applicable to polynomial and function space operators, enhancing stability and accuracy.

Findings

The new method yields numerically stable FSBP operators.

Operators constructed have higher accuracy at boundaries.

The approach outperforms traditional construction procedures.

Abstract

We introduce a novel construction procedure for one-dimensional summation-by-parts (SBP) operators. Existing construction procedures for FSBP operators of the form $D = P^{-1} Q$ proceed as follows: Given a boundary operator $B$, the norm matrix $P$ is first determined and then in a second step the complementary matrix $Q$ is calculated to finally get the FSBP operator $D$. In contrast, the approach proposed here determines the norm and complementary matrices, $P$ and $Q$, simultaneously by solving an optimization problem. The proposed construction procedure applies to classical SBP operators based on polynomial approximation and the broader class of function space SBP (FSBP) operators. According to our experiments, the presented approach yields a numerically stable construction procedure and FSBP operators with higher accuracy for diagonal norm difference operators at the boundaries than the traditional approach. Through numerical simulations, we highlight the advantages of our proposed technique.