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Tensor-Product Split-Simplex Summation-By-Parts Operators

Zelalem Arega Worku, Jason E. Hicken, David W. Zingg

Abstract

We present an approach to construct efficient sparse summation-by-parts (SBP) operators on triangles and tetrahedra with a tensor-product structure. The operators are constructed by splitting the simplices into quadrilateral or hexahedral subdomains, mapping tensor-product SBP operators onto the subdomains, and assembling back using a continuous-Galerkin-type procedure. These tensor-product split-simplex operators do not have repeated degrees of freedom at the interior interfaces between the split subdomains. Furthermore, they satisfy the SBP property by construction, leading to stable discretizations. The accuracy and sparsity of the operators substantially enhance the efficiency of SBP discretizations on simplicial meshes. The sparsity is particularly important for entropy-stable discretizations based on two-point flux functions, as it reduces the number of two-point flux computations. We demonstrate through numerical experiments that the operators exhibit efficiency surpassing that of the existing dense multidimensional SBP operators by more than an order of magnitude in many cases. This superiority is evident in both accuracy per degree of freedom and computational time required to achieve a specified error threshold.

Tensor-Product Split-Simplex Summation-By-Parts Operators

Abstract

We present an approach to construct efficient sparse summation-by-parts (SBP) operators on triangles and tetrahedra with a tensor-product structure. The operators are constructed by splitting the simplices into quadrilateral or hexahedral subdomains, mapping tensor-product SBP operators onto the subdomains, and assembling back using a continuous-Galerkin-type procedure. These tensor-product split-simplex operators do not have repeated degrees of freedom at the interior interfaces between the split subdomains. Furthermore, they satisfy the SBP property by construction, leading to stable discretizations. The accuracy and sparsity of the operators substantially enhance the efficiency of SBP discretizations on simplicial meshes. The sparsity is particularly important for entropy-stable discretizations based on two-point flux functions, as it reduces the number of two-point flux computations. We demonstrate through numerical experiments that the operators exhibit efficiency surpassing that of the existing dense multidimensional SBP operators by more than an order of magnitude in many cases. This superiority is evident in both accuracy per degree of freedom and computational time required to achieve a specified error threshold.
Paper Structure (17 sections, 3 theorems, 38 equations, 15 figures, 9 tables)

This paper contains 17 sections, 3 theorems, 38 equations, 15 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. SBP operators on curved elements
  5. Construction of TPSS operators
  6. Types of TPSS operators
  7. Accuracy and sparsity of TPSS operators
  8. Accuracy
  9. Sparsity
  10. Numerical results
  11. Advection problem
  12. Isentropic vortex problem
  13. Viscous Taylor--Green vortex problem
  14. Conclusions
  15. Grid convergence results for the advection problem
  16. ...and 2 more sections

Key Result

Theorem 1

Let ass:mapping hold and the metric terms be computed exactly, then for $\bm{u}_k\in\mathbb{R}^{n_p}$ holding the values of $\mathcal{U}\in\mathcal{C}^{p+1}(\hat{\Omega})$ at the nodes $S_{\Omega_{k}}$, the derivative operator given by eq:Dxik is order $p$ accurate, i.e.,

Figures (15)

  • Figure 1: Splitting the triangle and tetrahedron into three quadrilaterals and four hexahedra, respectively.
  • Figure 2: Quadrilateral and hexahedral reference elements and their node numbering.
  • Figure 3: Local and global node ordering for construction of the TPSS-SBP operator using the degree $p=1$ LGL one-dimensional operator. The global node ordering is shown in larger font size.
  • Figure 4: Examples of LGL and CSBP type TPSS operators. The degree $p$ TPSS operators are constructed using degree $p+d-1$ one-dimensional operators. The symbols $\mathbin{\ThisStyle{\vcenter{\hbox{$\SavedStyle\bullet$}}}}$ and $\textcolor{red}{\bm{\circ}}$ denote the collocated volume and facet nodes, respectively.
  • Figure 5: Grid convergence study of the $\mathsf{H}$-norm solution error for the 2D (left) and 3D (right) advection problems.
  • ...and 10 more figures

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Definition 2