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Multidimensional Summation-By-Parts Operators: General Theory and Application to Simplex Elements
Jason E. Hicken, David C. Del Rey Fernández, David W. Zingg
TL;DR
High-order SBP discretizations have largely been limited to tensor-product domains, which restricts the geometries considered in practice. This paper defines and analyzes multidimensional SBP operators, constructs triangle and tetrahedron examples, and reports accurate, time-stable linear-advection results on a doubly periodic domain.
Problem
Most SBP operators are one-dimensional and applied through tensor-product formulations, which complicate mesh generation and adaptation and limit geometric complexity.
Source: §1 · p. 1Method
The paper defines multidimensional SBP operators, analyzes diagonal-norm existence, constructs simplex operators, and develops weak boundary treatment and element assembly.
Source: §2.2 · p. 3, §6 · p. 17Results
Linear-advection results on a doubly periodic domain demonstrate the accuracy and time stability of the triangle-element SBP discretizations.
Source: §6 · pp. 17–18Takeaways & Limitations
Diagonal-norm SBP operators exist on a domain if and only if it admits positive-weight cubature with a full-rank polynomial-basis matrix at the cubature nodes.
Source: §6 · p. 17For simplex operators, the boundary operators are dense on boundary nodes, and the authors have not found sufficiently accurate diagonal boundary operators.
Source: §4.2 · p. 11
Abstract
from arXiv · showhide
Summation-by-parts (SBP) finite-difference discretizations share many attractive properties with Galerkin finite-element methods (FEMs), including time stability and superconvergent functionals; however, unlike FEMs, SBP operators are not completely determined by a basis, so the potential exists to tailor SBP operators to meet different objectives. To date, application of high-order SBP discretizations to multiple dimensions has been limited to tensor product domains. This paper presents a definition for multi-dimensional SBP finite-difference operators that is a natural extension of one-dimensional SBP operators. Theoretical implications of the definition are investigated for the special case of a diagonal norm (mass) matrix. In particular, a diagonal-norm SBP operator exists on a given domain if and only if there is a cubature rule with positive weights on that domain and the polynomial-basis matrix has full rank when evaluated at the cubature nodes. Appropriate simultaneous-approximation terms are developed to impose boundary conditions weakly, and the resulting discretizations are shown to be time stable. Concrete examples of multi-dimensional SBP operators are constructed for the triangle and tetrahedron; similarities and differences with spectral-element and spectral-difference methods are discussed. An assembly process is described that builds diagonal-norm SBP operators on a global domain from element-level operators. Numerical results of linear advection on a doubly periodic domain demonstrate the accuracy and time stability of the simplex operators.
1. Introduction.
Most SBP operators are one-dimensional and used through tensor-product grids, limiting practical geometric complexity; this work develops multidimensional operators for broader domains and grids. It defines and analyzes the operators, then constructs simplex examples.
1. Introduction.
- Tensor-product SBP formulations complicate mesh generation and adaptation and limit geometric complexity, motivating generalizations to unstructured grids.The paper describes two approaches: global operators on arbitrary node distributions, or reference-element operators assembled into a global discretization.Source: §1 · p. 1
- For high-order operators on arbitrary node distributions, stable diagonal-norm operators may require large stencils, and determining them generally requires a global solve.For some grids, no stable diagonal-norm operator exists for a given stencil width and design accuracy.Source: §1 · p. 2
- Reference-element SBP operators are a more common, less difficult route; prior work established dense-norm existence and a diagonal-norm spectral-element equivalence for Legendre-Gauss-Lobatto nodes.Carpenter and Gottlieb established dense-norm existence, while Gassner showed the stated diagonal-norm equivalence.Source: §1 · p. 2
- The paper defines multidimensional SBP operators on arbitrary grids and characterizes them theoretically, retaining properties such as discrete-divergence accuracy.The authors distinguish this objective from a candidate discrete-derivative operator that lacks some conventional SBP properties.Source: §1 · p. 2
- The paper constructs diagonal-norm SBP operators on triangles and tetrahedra, using freedom beyond a fixed polynomial basis to enforce SBP and obtain provably time-stable schemes.Diagonal norms are presented as attractive for explicit time marching and discretizations conserving nonquadratic invariants.Source: §1 · p. 2
2. Preliminaries.
The preliminaries establish the notation and define multidimensional SBP operators by accuracy and integration-by-parts-related conditions. They explain how the norm matrix and boundary terms support stability and the discrete analogue of integration by parts.
2. Preliminaries.
- The presentation focuses on two dimensions and the x-derivative, with higher-dimensional and y-direction results following directly; nodal vectors contain function values at the node set.The notation also uses sans-serif matrices for derivative operators and their entries and submatrices.Source: §2 · p. 3, §2.1 · p. 3
2.2. Multi-dimensional SBP operator definition.
- The degree-p SBP definition makes D_x approximate the first x-derivative and requires exact differentiation of polynomials through degree p.The definition includes a boundary relation involving the outward normal's x-component, with τ ≥ p.Source: §2.2 · pp. 3–4
- The monomial basis states the accuracy conditions concisely; SBP methods do not use a polynomial basis to define the solution as finite-element methods do.At least as many nodes as the polynomial basis size are necessary for the stated accuracy property.Source: §2.2 · p. 4
- A symmetric positive-definite norm matrix H supports a nonnegative discrete energy, while the SBP boundary relation supplies a discrete integration-by-parts analogue.The boundary term approximates the corresponding surface integral to order τ; in the diagonal-norm case, H also has a cubature-rule interpretation.Source: §2.2 · p. 4
3. Analysis of diagonal-norm, multi-dimensional, summation-by-parts operators.
The analysis characterizes when diagonal-norm multi-dimensional SBP operators exist and how their structure relates to cubature. It also shows how weak boundary enforcement yields time-stable advection discretizations under the stated conditions.
3. Analysis of diagonal-norm, multi-dimensional, summation-by-parts operators.
- A degree-p diagonal-norm SBP operator exists exactly when the nodes support a positive-weight degree-(2p−1) cubature and the evaluated polynomial-basis matrix has full column rank.The cubature condition is necessary for an existing operator and sufficient under the rank assumption.Source: §3 · p. 5
- The sufficiency proof sets H to the cubature weights and uses free matrix choices to satisfy the SBP conditions, with infinitely many operators possible for a given cubature rule.One stated construction takes Wx = 0, Gx = 0, and Fx = W^T H Vx.Source: §3 · pp. 6–7
- The theorem also characterizes Sx and Qx through the bilinear forms they approximate, a formulation useful for weak-form PDE discretizations.Source: §3 · p. 7
3.1. Stability Analysis.
- SAT penalties impose boundary conditions weakly, and the boundary-operator decomposition needed for the stability argument exists when Fx = Fy = 0.The boundary data vector must accurately reconstruct the prescribed boundary data.Source: §3.1 · p. 8
- For constant-coefficient advection with homogeneous boundary conditions and bounded initial data, the SBP semi-discretization is time stable.Source: §3.1 · p. 8
4. Constructing the operators.
The section constructs diagonal-norm SBP operators for triangles and tetrahedra, then shows how element-level operators yield a degree-p SBP operator on the global node set. Its construction uses positive-weight cubature, boundary operators, and accuracy conditions for the antisymmetric part.
4.1. The node coordinates and the norm matrix.
- Diagonal entries of H are positive cubature weights exact for polynomials of total degree 2p − 1.The triangle rules are adopted for degrees p = 1, 2, 3, and 4; tetrahedral cubature parameters and weights are found by solving nonlinear accuracy conditions.Source: §4.1 · p. 9
4.2. The boundary operators.
- The constructed boundary operators use nodes on each facet, although the SBP definition itself does not require boundary nodes.For the simplex operators considered, boundary operators restricted to boundary nodes are dense; the authors report no sufficiently accurate diagonal alternative.Source: §4.1 · p. 9, §4.2 · p. 11
4.3. The antisymmetric part.
- Orthogonal polynomial bases avoid ill-conditioned monomial Vandermonde matrices when imposing accuracy conditions on Qx.For p ≥ 3, the compatible system has infinitely many solutions, so the authors choose a minimum-norm least-squares solution.Source: §4.3 · pp. 11–12
4.4. Similarities and differences with existing operators.
- Higher-order diagonal-mass spectral-element operators do not satisfy the SBP definition for p ≥ 2 and are unstable without filtering or numerical dissipation.Making the relevant matrix satisfy the definition with a cubature exact for the cardinal basis would require additional points and sacrifice collocation efficiency.Source: §4.4 · p. 12
- Compared with spectral-difference methods, collocated SBP operators use more unknowns for a given accuracy but eliminate reconstruction.The unknown-count increase applies to discontinuous methods; assembling a global SBP operator can significantly reduce the number of unknowns.Source: §4.4 · p. 12
4.5. Assembly of global SBP operators from elemental operators.
- Assembled element operators yield a degree-p SBP operator on the global node set.The assembly uses shared global indices for local nodes, and adjacent-element boundary fluxes cancel analytically.Source: §4.5 · pp. 13–14
5. Results.
The study evaluates continuous and discontinuous triangle-element SBP discretizations for periodic linear advection, examining accuracy, efficiency, and stability. Accuracy depends on the discretization and degree, while the SBP operators remain stable in the reported test.
5. Results.
- The tests use a nonuniform triangular mesh, affine element mappings, periodic boundaries, and one-period runs with a CFL number of 0.9CFLmax for the accuracy study.The mesh spans N = 4 to N = 64; the study uses both C-SBP and D-SBP discretizations.Source: §5 · pp. 14–15, §5.1 · p. 15
5.1. Accuracy and efficiency studies.
- D-SBP and odd-order C-SBP discretizations converge at O(h^(p+1)), whereas even-order C-SBP operators of p = 2 and p = 4 converge at O(h^p).The even-order continuous operators exhibit checkerboarding; the p = 1 spatial error is smooth while the p = 2 error is oscillatory.Source: §5.1 · p. 15, Figure 6 · p. 16, Figure 5 · p. 16
- High-order operators show efficiency in both continuous and discontinuous discretizations in the normalized L2 error-versus-CPU-time comparison.The runs used Julia 0.4.0 on an Intel Core i5-3570K processor.Source: §5.1 · p. 16, Figure 8 · p. 17, Figure 7 · p. 17
5.2. Stability studies.
- For the periodic problem, C-SBP eigenvalues are imaginary to machine precision; high-order spectral-element operators have a real component.The p = 1 spectral-element operator also has a purely imaginary spectrum because it is identical to the linear SBP operator.Source: §5.2 · pp. 16–17
- The SBP operators are stable for the tested linear advection problem, while the spectral-element operators are unstable.The small linear decrease in SBP energy error is attributed to temporal errors and can be reduced with a different time integrator or smaller CFL number.Source: §5.2 · p. 17
6. Conclusions.
The paper establishes a general multidimensional SBP framework, constructs diagonal-norm operators for simplices, and tests triangle-element operators on periodic linear advection. The reported results demonstrate accuracy and time stability in that setting.
6. Conclusions.
- For a domain, diagonal-norm SBP operators exist if and only if a positive-weight cubature rule exists and its evaluated polynomial-basis matrix has full rank.The paper also develops SATs for weak boundary-condition imposition and shows that the resulting SBP-SAT linear-advection discretization is time stable.Source: §6 · p. 17
- The paper constructs diagonal-norm SBP operators for triangles and tetrahedra and presents an assembly procedure for building global operators from element-wise operators.These are reported as the first SBP operators of degree p ≥ 2 on these domains.Source: §6 · p. 17
- Triangle-element operators tested with continuous and discontinuous coupling demonstrate accuracy and time stability for linear advection on a doubly periodic domain.The authors suggest the operators could be effective for long-time turbulent-flow simulations on complex domains.Source: §6 · pp. 17–18