Why summation by parts is not enough

TL;DR

SBP operators defined by the discrete relation $D = P^{-1}Q$ with $Q+Q^T = B$ are not automatically accurate for hyperbolic PDEs; unresolved modes and nullspace issues can degrade performance despite conservation. The paper introduces two design strategies within an optimization-based FSBP framework on arbitrary grids: enforcing sparsity to promote favorable nullspace properties and spectral behavior, and regularizing to enlarge the exactness basis with an augmented set $ mathcal{G}$ while preserving SBP constraints. The key contributions include formalizing nullspace consistency as essential for convergence, demonstrating the benefits of sparse-structured FSBP operators, and showing that regularized augmentation yields numerically nullspace-consistent operators with superior accuracy in 1D advection and 2D Euler tests. Overall, the work provides a flexible, general framework for high-order, structure-preserving discretizations applicable to general grids and potentially meshfree methods, with implications for robust simulations of hyperbolic conservation laws.

Abstract

We investigate the construction and performance of summation-by-parts (SBP) operators, which offer a powerful framework for the systematic development of structure-preserving numerical discretizations of partial differential equations. Previous approaches for the construction of SBP operators have usually relied on either local methods or sparse differentiation matrices, as commonly used in finite difference schemes. However, these methods often impose implicit requirements that are not part of the formal SBP definition. We demonstrate that adherence to the SBP definition alone does not guarantee the desired accuracy, and we identify conditions for SBP operators to achieve both accuracy and stability. Specifically, we analyze the error minimization for an augmented basis, discuss the role of sparsity, and examine the importance of nullspace consistency in the construction of SBP operators. Furthermore, we show how these design criteria can be integrated into a recently proposed optimization-based construction procedure for function space SBP (FSBP) operators on arbitrary grids. Our findings are supported by numerical experiments that illustrate the improved accuracy for the numerical solution using the proposed SBP operators.

Figures (5)

• Figure 1: Numerical solutions of the linear advection equation using FSBP operators with different polynomial degrees $d$ on $N=50$ nodes and at $t=1.75$. The left subplot shows the solution for $k=1$ and the right subplot for $k=2$ in the initial condition $u_0(x) = \sin(k\pi x)$.
• Figure 1: $L^2$- and $L^\infty$-errors for the one-dimensional linear advection equation using FSBP operators based on $\mathcal{P}_3 = \mathop{\mathrm{span}}\nolimits\{1, x, x^2, x^3\}$ and $\mathcal{T} = \mathop{\mathrm{span}}\nolimits\{1, x, \sin(\pi x), \cos(\pi x)\}$ on $8$ blocks with $N=15$ nodes per block. Sparse operators with bandwidth $b = 3$ are compared to dense operators.
• Figure 2: $L^2$- and $L^\infty$-errors summed across all variables for the two-dimensional compressible Euler equations using FSBP operators based on $\mathcal{P}_3 = \mathop{\mathrm{span}}\nolimits\{1, x, x^2, x^3\}$ and $\mathcal{T} = \mathop{\mathrm{span}}\nolimits\{1, x, \sin(\pi x), \cos(\pi x)\}$ with global operators on $N=50$ equidistant nodes. Sparse operators with bandwidth $b = 3$ are compared to dense operators.
• Figure 3: $L^2$- and $L^\infty$-errors for the one-dimensional linear advection equation using FSBP operators based on $\mathcal{P}_3 = \mathop{\mathrm{span}}\nolimits\{1, x, x^2, x^3\}$ without regularization and with regularization on $\mathcal{G} = \{\sin(\pi x), \cos(\pi x)\}$ on $8$ blocks with $N=15$ nodes per block.
• Figure 4: $L^2$- and $L^\infty$-errors for the two-dimensional compressible Euler equations using FSBP operators based on $\mathcal{P}_3 = \mathop{\mathrm{span}}\nolimits\{1, x, x^2, x^3\}$ without regularization and with regularization on $\mathcal{G} = \{\sin(\pi x), \cos(\pi x)\}$ on $8$ blocks with $N=15$ nodes per block.

Theorems & Definitions (10)

• Definition 2.1: FSBP operators
• Remark 2.2
• Example 3.1
• Lemma 4.1
• Proof 1
• Lemma 4.2
• Proof 2
• Remark 4.3
• Remark 4.4
• Remark 4.5