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

TL;DR

This paper introduces tensor-product split-simplex SBP operators that are sparse, efficient, and stable for discretizations on simplicial meshes, significantly outperforming existing dense operators in accuracy and computational efficiency.

Contribution

The authors develop a novel tensor-product split-simplex SBP operator construction that enhances efficiency and stability for simplicial mesh discretizations, with superior performance over existing methods.

Findings

Operators are more than ten times more efficient than dense counterparts.

Achieve comparable accuracy with fewer degrees of freedom.

Numerical experiments confirm improved computational time and accuracy.

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.