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Efficient Entropy-Stable Discontinuous Spectral-Element Methods Using Tensor-Product Summation-by-Parts Operators on Triangles and Tetrahedra

Tristan Montoya, David W. Zingg

TL;DR

This work develops high-order entropy-stable discontinuous spectral-element methods on curved triangular and tetrahedral meshes by leveraging sparse tensor-product SBP operators in collapsed coordinates combined with a Proriol–Koornwinder–Dubiner modal basis. A weight-adjusted mass-inverse preserves tensor-product structure, enabling ${O(p^{d+1})}$ local complexity and efficient sum-factorization, while a conservative curl-based metric-term approximation ensures discrete metric identities and free-stream preservation. The entropy-stable flux-differencing framework uses entropy-conservative two-point fluxes and a dissipative interface flux to guarantee semi-discrete entropy bounds, with a hybridized SBP reformulation underpinning conservation, free-stream preservation, and stability proofs. Numerical experiments with the compressible Euler equations on curvilinear triangles/tetrahedra confirm conservation, entropy behavior, and comparable accuracy to multidimensional SBP schemes, while demonstrating robustness in under-resolved regimes. The results indicate that tensor-product SBP operators extend entropy stability to higher polynomial degrees on simplicial elements, offering substantial computational advantages for high-order simulations on complex geometries.

Abstract

We present a new class of efficient and robust discontinuous spectral-element methods of arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular and tetrahedral unstructured grids. Such discretizations employ a recently introduced family of sparse tensor-product summation-by-parts (SBP) operators in collapsed coordinates within an entropy-stable modal formulation. The proposed algorithms exploit the structure of such SBP operators alongside that of the Proriol-Koornwinder-Dubiner polynomial basis, and a weight-adjusted approximation is used to efficiently invert the local mass matrix for curvilinear elements. Using such techniques, the number of required entropy-conservative two-point flux evaluations between pairs of quadrature nodes is significantly reduced relative to existing entropy-stable formulations using (non-tensor-product) multidimensional SBP operators, particularly for high polynomial degrees, with an improvement in time complexity from $\mathcal{O}(p^{2d})$ to $\mathcal{O}(p^{d+1})$, where $p$ is the polynomial degree of the approximation and $d$ is the number of spatial dimensions. In numerical experiments involving smooth solutions to the compressible Euler equations, the proposed tensor-product schemes demonstrate similar levels of accuracy for a given mesh and polynomial degree to those using multidimensional SBP operators based on symmetric quadrature rules. Furthermore, both operator families are shown to give rise to entropy-stable methods which exhibit excellent robustness for under-resolved problems. Such results suggest that the algorithmic advantages resulting from the use of tensor-product operators are obtained without compromising accuracy or robustness, enabling the efficient extension of the benefits of entropy stability to higher polynomial degrees than previously considered for triangular and tetrahedral elements.

Efficient Entropy-Stable Discontinuous Spectral-Element Methods Using Tensor-Product Summation-by-Parts Operators on Triangles and Tetrahedra

TL;DR

This work develops high-order entropy-stable discontinuous spectral-element methods on curved triangular and tetrahedral meshes by leveraging sparse tensor-product SBP operators in collapsed coordinates combined with a Proriol–Koornwinder–Dubiner modal basis. A weight-adjusted mass-inverse preserves tensor-product structure, enabling local complexity and efficient sum-factorization, while a conservative curl-based metric-term approximation ensures discrete metric identities and free-stream preservation. The entropy-stable flux-differencing framework uses entropy-conservative two-point fluxes and a dissipative interface flux to guarantee semi-discrete entropy bounds, with a hybridized SBP reformulation underpinning conservation, free-stream preservation, and stability proofs. Numerical experiments with the compressible Euler equations on curvilinear triangles/tetrahedra confirm conservation, entropy behavior, and comparable accuracy to multidimensional SBP schemes, while demonstrating robustness in under-resolved regimes. The results indicate that tensor-product SBP operators extend entropy stability to higher polynomial degrees on simplicial elements, offering substantial computational advantages for high-order simulations on complex geometries.

Abstract

We present a new class of efficient and robust discontinuous spectral-element methods of arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular and tetrahedral unstructured grids. Such discretizations employ a recently introduced family of sparse tensor-product summation-by-parts (SBP) operators in collapsed coordinates within an entropy-stable modal formulation. The proposed algorithms exploit the structure of such SBP operators alongside that of the Proriol-Koornwinder-Dubiner polynomial basis, and a weight-adjusted approximation is used to efficiently invert the local mass matrix for curvilinear elements. Using such techniques, the number of required entropy-conservative two-point flux evaluations between pairs of quadrature nodes is significantly reduced relative to existing entropy-stable formulations using (non-tensor-product) multidimensional SBP operators, particularly for high polynomial degrees, with an improvement in time complexity from to , where is the polynomial degree of the approximation and is the number of spatial dimensions. In numerical experiments involving smooth solutions to the compressible Euler equations, the proposed tensor-product schemes demonstrate similar levels of accuracy for a given mesh and polynomial degree to those using multidimensional SBP operators based on symmetric quadrature rules. Furthermore, both operator families are shown to give rise to entropy-stable methods which exhibit excellent robustness for under-resolved problems. Such results suggest that the algorithmic advantages resulting from the use of tensor-product operators are obtained without compromising accuracy or robustness, enabling the efficient extension of the benefits of entropy stability to higher polynomial degrees than previously considered for triangular and tetrahedral elements.
Paper Structure (29 sections, 7 theorems, 116 equations, 9 figures)

This paper contains 29 sections, 7 theorems, 116 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Systems of conservation laws and the entropy inequality
  5. Summation-by-parts operators
  6. Decomposition of the boundary operators
  7. Orthogonal polynomials and Gaussian quadrature rules
  8. Tensor-product summation-by-parts operators in collapsed coordinates
  9. Tensor-product summation-by-parts operators on the reference triangle
  10. Tensor-product summation-by-parts operators on the reference tetrahedron
  11. Entropy-stable discontinuous spectral-element framework
  12. Mesh and coordinate transformation
  13. Approximation of the metric terms
  14. Modal polynomial expansions on triangles and tetrahedra
  15. Discontinuous spectral-element methods using summation-by-parts operators
  16. ...and 14 more sections

Key Result

Lemma 1

Given any nodal SBP operator $\underline{\underline{D}}_{}{}^{(l)} \coloneqq \underline{\underline{W}}_{}{}^{-1}(\underline{\underline{S}}_{}{}^{(l)} + \tfrac{1}{2}\underline{\underline{E}}_{}{}^{(l)})$ on the reference element in the sense of def:sbp for which $\underline{\underline{E}}_{}{}^{(l)}$ satisfying the following "SBP-like" property from chan_discretely_entropy_conservative_dg_sbp_18:

Figures (9)

  • Figure 1: Illustration of the collapsed coordinate transformation $\boldsymbol{\xi} = \boldsymbol{\chi}(\boldsymbol{\eta})$ from the square to the reference triangle
  • Figure 2: Illustration of the collapsed coordinate transformation $\boldsymbol{\xi} = \boldsymbol{\chi}(\boldsymbol{\eta})$ from the cube to the reference tetrahedron
  • Figure 3: Tensor-product volume quadrature nodes for SBP operators of degree $q = 4$ on the triangle and tetrahedron
  • Figure 4: Symmetric quadrature nodes for multidimensional SBP operators of degree $q = 4$ on the triangle and tetrahedron
  • Figure 5: Number of entropy-conservative two-point flux evaluations in local flux-differencing terms
  • ...and 4 more figures

Theorems & Definitions (30)

  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Definition 3
  • ...and 20 more