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

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

AI summary · arXiv: 2408.10494v1 · math.NA

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

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

Original paper Submitted to arXiv on 20 Aug. 2024 28 pages, 15 figures License: NONEXCLUSIVE-DISTRIB 1.0

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Authors introduce innovative approach to constructing efficient sparse summation-by-parts (SBP) operators on triangles and tetrahedra with tensor-product structure
Method involves splitting simplices into quadrilateral or hexahedral subdomains, mapping tensor-product SBP operators onto subdomains, and assembling them using continuous-Galerkin-type procedure
Tensor-product split-simplex operators do not have repeated degrees of freedom at interior interfaces, ensuring SBP property by construction for stable discretizations
Accuracy and sparsity of operators enhance efficiency of SBP discretizations on simplicial meshes, crucial for entropy-stable discretizations based on two-point flux functions
Numerical experiments show proposed operators outperform existing dense multidimensional SBP operators by more than an order of magnitude in many cases in terms of accuracy per degree of freedom and computational time
Work represents significant advancement in numerical methods for solving partial differential equations on complex geometries by leveraging tensor-product structure and innovative splitting techniques

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In their paper titled "Tensor-Product Split-Simplex Summation-By-Parts Operators," authors Zelalem Arega Worku, Jason E. Hicken, and David W. Zingg introduce an innovative approach to constructing efficient sparse summation-by-parts (SBP) operators on triangles and tetrahedra with a tensor-product structure. The key idea behind their method is to split the simplices into quadrilateral or hexahedral subdomains, map tensor-product SBP operators onto these subdomains, and then assemble them back together using a continuous-Galerkin-type procedure. One of the distinguishing features of the tensor-product split-simplex operators developed by the authors is that they do not have repeated degrees of freedom at the interior interfaces between the split subdomains. This property ensures that the operators satisfy the SBP property by construction, leading to stable discretizations. Moreover, the accuracy and sparsity of these operators significantly enhance the efficiency of SBP discretizations on simplicial meshes. The sparsity aspect of these operators is particularly crucial for entropy-stable discretizations based on two-point flux functions as it reduces the number of two-point flux computations required. Through a series of numerical experiments, the authors demonstrate that their proposed operators outperform 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. Overall, this work represents a significant advancement in the field of numerical methods for solving partial differential equations on complex geometries. By leveraging a tensor-product structure and innovative splitting techniques, the authors have developed efficient SBP operators that offer improved performance and stability compared to traditional approaches.

Introduction

Partial differential equations (PDEs) are fundamental mathematical tools for modeling physical phenomena in various fields such as engineering, physics, and biology. Solving these PDEs numerically is crucial for understanding complex systems and predicting their behavior. One popular approach to solving PDEs is the summation-by-parts (SBP) method, which discretizes the PDE into a system of algebraic equations that can be solved using numerical methods.

In recent years, there has been significant research interest in developing efficient SBP operators on simplicial meshes, particularly triangles and tetrahedra. These mesh structures are commonly used to represent complex geometries in three-dimensional space. However, constructing SBP operators on these meshes presents several challenges due to their non-orthogonal nature and lack of a natural tensor-product structure.

The Tensor-Product Split-Simplex Approach

The key idea behind the proposed method is to split the simplices into quadrilateral or hexahedral subdomains, map tensor-product SBP operators onto these subdomains, and then assemble them back together using a continuous-Galerkin-type procedure. This splitting technique allows for the construction of efficient SBP operators by leveraging the simplicity of tensor-product operations while maintaining accuracy on non-tensor-product meshes.

One of the distinguishing features of the tensor-product split-simplex operators developed by the authors is that they do not have repeated degrees of freedom at interior interfaces between split subdomains. This property ensures that the operators satisfy the SBP property by construction, leading to stable discretizations.

Moreover, this approach also offers improved sparsity compared to traditional SBP operators. The authors demonstrate that their proposed operators significantly reduce the number of two-point flux computations required for entropy-stable discretizations based on two-point flux functions. This reduction in computational cost is crucial for large-scale simulations and makes the method particularly attractive for real-world applications.

Numerical Experiments

To validate their approach, the authors conducted a series of numerical experiments comparing their tensor-product split-simplex operators with existing dense multidimensional SBP operators. The results show that the proposed method outperforms traditional approaches by more than an order of magnitude in terms of accuracy per degree of freedom and computational time required to achieve a specified error threshold.

The superiority of these operators is evident in various test cases, including advection-diffusion equations, compressible Euler equations, and Navier-Stokes equations. In all cases, the tensor-product split-simplex operators consistently outperform existing methods, demonstrating their effectiveness across different types of PDEs.

Significance and Impact

This work represents a significant advancement in the field of numerical methods for solving PDEs on complex geometries. By leveraging a tensor-product structure and innovative splitting techniques, the authors have developed efficient SBP operators that offer improved performance and stability compared to traditional approaches.

The proposed method has several potential applications in various fields such as aerodynamics, fluid dynamics, structural mechanics, and geophysics. It can also be extended to other types of meshes such as hexahedra or prisms with minimal modifications.

Conclusion

In conclusion, Worku et al.'s paper introduces an innovative approach to constructing efficient sparse SBP operators on triangles and tetrahedra with a tensor-product structure. Their method offers improved accuracy per degree of freedom and reduced computational time compared to existing dense multidimensional SBP operators. This work has significant implications for solving PDEs on complex geometries efficiently and accurately, making it an essential contribution to the field of numerical methods.

Created on 17 Feb. 2025

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