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

TL;DR

This work tackles the challenge of constructing high-order function-space SBP operators on arbitrary grids by reframing the problem as an optimization that simultaneously determines the norm matrix $P$ and the differentiation matrix $D$ via $D = P^{-1}Q$. By formulating the operator as a constrained quadratic minimization over $X=[S,P]$ with $S$ anti-symmetric and $P$ diagonal, and then removing constraints through a parameterization that enforces positivity and constant-sum of $P$, the authors achieve numerically stable, high-accuracy FSBP operators without requiring an explicit $(\mathcal{F}^2)'$-exact quadrature. Conditioning is improved by adopting discrete Sobolev orthogonality for the basis, ensuring a well-posed optimization problem, while the method supports arbitrary grids and non-polynomial spaces, as demonstrated on linear advection and Schrödinger equations. The results indicate enhanced boundary accuracy, robustness to grid choices, and potential for extensions to RBF discretizations and neural-network–based approaches on complex domains.

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.

Figures (8)

• Figure 1: Different sigmoid functions with the characteristic 'S-shape' plotted between $[-10,10]$.
• Figure 1: Convergence Analysis for our new operators. The errors of the solution converge with eight order in the energy norm induced by the quadrature in $P$.
• Figure 2: Errors for the SBP operators with and without \ref{['eqparamP']}
• Figure 3: Errors for SBP operators on equidistant and random points
• Figure 4: Convergence Analysis for our new operators. The errors of the solution converge in the energy norm induced by the quadrature in $P$. The $y$ axis denotes the error, and the $x$ axis refers to the refinement of the grid.

Theorems & Definitions (9)

• Definition 2.1: FSBP operators
• Remark 2.2: Limitation in polynomial SBP theory
• Remark 2.4
• Lemma 3.1
• Proof 1
• Lemma 3.2
• Remark 3.3
• Remark 3.4: Solving the optimization problem
• Remark 4.1