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Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra

Tristan Montoya, David W. Zingg

TL;DR

The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms.

Abstract

We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.

Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra

TL;DR

The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms.

Abstract

We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.
Paper Structure (33 sections, 4 theorems, 82 equations, 5 figures)

This paper contains 33 sections, 4 theorems, 82 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Summation-by-parts operators
  5. Decomposition of the boundary operators
  6. Curvilinear meshes
  7. Approximation on the physical element
  8. Tensor-product approximations on the reference triangle
  9. Nodal sets and interpolants
  10. Volume quadrature
  11. Facet quadrature
  12. Summation-by-parts operators
  13. Tensor-product approximations on the reference tetrahedron
  14. Nodal sets and interpolants
  15. Volume quadrature
  16. ...and 18 more sections

Key Result

Lemma 3.1

\newlabellem:accuracy_2d0 The tensor-product interpolation operator defined in eq:nodal_tensor is exact for any polynomial $V \in \mathbb{P}_q(\hat{\Omega})$, satisfying $(\mathcal{I}_q V)(\boldsymbol{\xi}) = V(\boldsymbol{\xi})$ for all $\boldsymbol{\xi} \in \hat{\Omega} \setminus \{[-1,1]^\mathr

Figures (5)

  • Figure 1: Illustration of the collapsed coordinate transformation $\boldsymbol{\xi} = \boldsymbol{\chi}(\boldsymbol{\eta})$ from the square to the reference triangle (top) and from the cube to the reference tetrahedron (bottom)
  • Figure 1: Time evolution of the conservation and energy residuals for tensor-product discretizations on triangles (top row) and tetrahedra (bottom row) plotted at 101 equispaced snapshots
  • Figure 2: Variation in spectral radius of the semi-discrete advection operator with polynomial degree for discretizations on triangles (top row) and tetrahedra (bottom row); solid and dashed lines denote upwind and central numerical fluxes, respectively
  • Figure 3: Convergence with respect to $h$ and $p$ for discretizations of the linear advection equation on triangles (top row) and tetrahedra (bottom row); solid and dashed lines denote upwind and central numerical fluxes, respectively
  • Figure 4: Floating-point operation count for local time derivative evaluation on triangles (left) and tetrahedra (right); solid and dashed lines denote reference-operator and physical-operator algorithms, respectively

Theorems & Definitions (12)

  • Definition 2.1: Nodal SBP operator
  • Lemma 3.1
  • Proof 1
  • Theorem 3.2
  • Proof 2
  • Remark 3.3
  • Lemma 4.1
  • Proof 3
  • Remark 4.2
  • Theorem 4.3
  • ...and 2 more