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Summation-By-Parts Operators and High-Order Quadrature
Jason E. Hicken, David W. Zingg
TL;DR
On uniform grids, the quadrature rule for finite-difference methods must do more than approximate integrals accurately: it may also need to preserve functional properties such as the divergence theorem. This paper relates SBP weight matrices to end-corrected trapezoid rules and establishes accuracy results and implications for SBP discretizations.
Problem
Method
The paper relates SBP weight matrices to trapezoid rules with end corrections and analyzes the resulting quadrature accuracy.
Source: §3.1 · p. 7, §5 · p. 17Results
O(h^2s) accurate approximation of the L2 norm is provided by the SBP energy norm, and multidimensional SBP discretizations mimic the divergence theorem to order h^2s on specified curvilinear domains.
Source: §3.3 · p. 11, §5 · p. 17Takeaways & Limitations
Multidimensional SBP quadrature applied to the discrete divergence produces an accurate boundary-flux quadrature with no interior points involved.
Source: §5 · p. 17The leading error term for SBP quadrature is unknown beyond its asymptotic form, so the paper makes no general claim about relative accuracy among weight matrices.
Source: §5 · p. 17
Abstract
from arXiv · showhide
Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. This property can be useful in constructing energy-stable discretizations of partial differential vequations. SBP operators are defined by a weight matrix and a difference operator, with the latter designed to approximate $d/dx$ to a specified order of accuracy. The accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. The accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; for example, the discrete norm accurately approximates the $L^{2}$ norm for functions, and multi-dimensional SBP discretizations accurately mimic the divergence theorem.
1. Introduction
On uniform finite-difference grids, selecting a quadrature rule is less straightforward than choosing Gaussian quadrature when weights and nodes are free. The paper examines SBP weight-matrix quadrature, relating it to end-corrected trapezoid rules and showing how it can preserve the discrete divergence theorem.
1. Introduction
- SBP weight matrices induce high-order quadrature, a surprising property because quadrature accuracy is not explicitly part of the SBP definition.The authors report that this relationship had not previously been discussed in the literature.Source: §1 · p. 2
- SBP operators have been used to construct linearly time-stable discretizations for PDEs including the Euler, Navier-Stokes, and Einstein equations [2, 3],, [7].The paper investigates quadrature particularly suited to high-order SBP finite-difference methods.Source: §1 · pp. 1–2
- SBP quadrature uses the operator's weight matrix, offering a way for functional estimates to preserve properties of the true functional beyond accuracy.The introduction contrasts this motivation with available composite Newton-Cotes and Gregory-type rules.Source: §1 · p. 2
- Classical quadrature rules can accurately integrate uniformly spaced data yet typically fail to make arbitrary high-order finite-difference divergence approximations mimic the divergence theorem.The paper defines this mimicry as accuracy together with volume quadrature producing surface quadrature.Source: §1 · p. 2
- The paper derives conditions for end-corrected trapezoid-rule weights, studies coordinate transformations, and verifies the results with numerical examples.It also examines the implications for SBP discretizations of PDEs.Source: §1 · p. 2
2. Notation and definitions
The section sets notation for SBP operators on a uniform grid and describes the weight matrices and accuracy properties relevant to the paper's quadrature analysis. It also identifies the accuracy question left open by the SBP integration-by-parts property and distinguishes diagonal norms from full and restricted-full norms.
2. Notation and definitions
- The grid partitions [0, 1] into n+1 evenly spaced nodes x_v=vh with spacing h=1/n.Other finite intervals and nonuniform spacing can be handled with an appropriate mapping.Source: §2 · p. 2
- An SBP operator uses a symmetric-positive-definite weight matrix H and a difference operator D=H^-1Q, with Q+Q^T equal to the endpoint boundary matrix.The defining property is the discrete analogue of integration by parts.Source: §2 · pp. 2–3
- For a 2s-order SBP operator, the derivative approximation has order 2s at internal nodes, while boundary-node stencil accuracy can differ.The even internal order follows from using centered-difference schemes.Source: §2 · p. 3
- Although the SBP property leads to energy-stable PDE discretizations, its analogy with integration by parts does not by itself establish an accurate discretization.The section frames accuracy of that discrete identity as an open question for the analysis.Source: §2 · p. 3
- Full-norm SBP operators can attain boundary accuracy 2s−1 with r=2s, and restricted-full norms can maintain that boundary accuracy.The full-norm result is attributed to Kreiss and Scherer, and the restricted-full result to Strand.Source: §2 · p. 4
- Diagonal-norm operators support provably stable PDE discretizations on curvilinear grids but have boundary accuracy s when internal accuracy is 2s.For hyperbolic systems, the stated solution accuracy is limited to order s+1; functionals from dual-consistent diagonal-norm discretizations can be 2s-order accurate.Source: §2 · p. 4
3. Theory
The theory establishes that SBP weight matrices provide high-order quadrature, then extends that accuracy to curvilinear domains under a matching-Jacobian condition. It also shows that SBP discretizations mimic the divergence theorem on curvilinear domains.
3.1. One-dimensional SBP Quadrature
- Theorem 1 gives necessary and sufficient coefficient conditions for q-order trapezoid rules with end corrections, with extra degrees of freedom when q−1<r.Source: §3.1 · pp. 5, 7
- SBP weight matrices yield 2s-order-accurate quadrature for integrands in C2s when the associated first-derivative operator is 2s-order accurate internally.Source: §3.1 · pp. 7–8
3.2. SBP Quadrature and Coordinate Transformations
- On curvilinear domains, the discrete SBP product remains 2s-order accurate when the Jacobian uses the same SBP derivative as the norm, assuming sufficient smoothness.Source: §3.2 · pp. 8–10
3.3. SBP Operators and the Divergence Theorem
- On curvilinear domains diffeomorphic to the d-cube, SBP discretizations mimic the d-dimensional divergence theorem to order h^2s, with an accurate discrete result depending only on boundary terms.Source: §3.3 · pp. 11–12
4. Examples
Across the examples, SBP quadrature attains its expected 2s-order accuracy in one-dimensional and curvilinear settings. For integrated divergence, it also depends only on boundary nodes and approximates surface flux at 2s order.
4.1. One-dimensional Quadrature
- All tested one-dimensional rules converge at the expected 2s asymptotic rate; for n ≥64, diag-2-4 is almost an order of magnitude more accurate than full-3-4.The comparison concerns absolute errors and does not establish a general ranking of schemes.Source: §4.1 · p. 13, Table 2 · p. 13, Figure 1 · p. 14
4.2. Multi-dimensional Quadrature on a Curvilinear Domain
- Curvilinear quadrature remains 2s-order accurate when the same diagonal-norm SBP operator approximates the Jacobian and quadrature; mixing diag-3-6 and diag-2-4 yields only order 3.The mixed scheme uses diag-3-6 for Jacobian derivatives and diag-2-4 for quadrature, and is less accurate than using diag-2-4 for both.Source: §4.2 · pp. 13, 15, Table 3 · p. 15
4.3. Discrete Divergence Theorem
- The integrated SBP divergence is 2s-order accurate and depends only on boundary nodes, with the boundary expression matching the result to machine error.The boundary-node result is a 2s-order approximation to surface flux.Source: §4.3 · pp. 16–17, Table 4 · p. 17
5. Discussion and Conclusions
SBP weight matrices are related to trapezoid rules with end corrections, and the paper identifies consequences for norms, functionals, and multidimensional divergence calculations. The authors do not claim these quadrature rules are optimal, and the leading error term remains unknown.
5. Discussion and Conclusions
- SBP weight matrices are related to trapezoid rules with end corrections, but their optimality over uniformly spaced grids is not established.The leading error term is also unknown, preventing general comparisons of the relative accuracy of different weight matrices.Source: §5 · p. 17
- The SBP energy norm approximates the L2 norm for functions on [0, 1] with O(h^2s) accuracy.This norm is frequently used in stability analyses of SBP-based PDE discretizations.Source: §5 · p. 17
- Multidimensional SBP discretizations using Kronecker products mimic the divergence theorem, yielding accurate boundary-flux quadrature without interior points.The weight-matrix quadrature is applied to the discrete divergence.Source: §5 · p. 17
- Diagonal-norm SBP operators with 2s-order interior schemes have s-order-accurate boundary closures, limiting PDE solutions to s + 1 order accuracy.For dual-consistent SBP discretizations, using the corresponding quadrature to calculate functionals yields superconvergent 2s-order-accurate functionals (see also).Source: §5 · p. 17
- The SBP weight matrix appears to be the natural quadrature rule for functionals evaluated from corresponding SBP discretizations.The authors cite the rule's compatibility with the properties of the true functional as its value, rather than claiming optimality against existing schemes.Source: §5 · p. 17