Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Quadrature Rules on Triangles and Tetrahedra for Multidimensional Summation-By-Parts Operators (2311.15576v1)

Published 27 Nov 2023 in math.NA and cs.NA

Abstract: Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal-$ \mathsf{E} $ operators, are attractive for entropy-stable discretizations from an efficiency standpoint. However, there is a limited number of such operators, and those currently in existence often have a relatively high node count for a given polynomial order due to a scarcity of suitable quadrature rules. We present several new symmetric positive-weight quadrature rules on triangles and tetrahedra that are suitable for construction of diagonal-$ \mathsf{E} $ SBP operators. For triangles, quadrature rules of degree one through twenty with facet nodes that correspond to the Legendre-Gauss-Lobatto (LGL) and Legendre-Gauss (LG) quadrature rules are derived. For tetrahedra, quadrature rules of degree one through ten are presented along with the corresponding facet quadrature rules. All of the quadrature rules are provided in a supplementary data repository. The quadrature rules are used to construct novel SBP diagonal-$ \mathsf{E} $ operators, whose accuracy and maximum timestep restrictions are studied numerically.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (38)
  1. Chan, J.: On discretely entropy conservative and entropy stable discontinuous Galerkin methods. Journal of Computational Physics 362, 346–374 (2018)
  2. Journal of Computational Physics 345, 427–461 (2017)
  3. SIAM Journal on Numerical Analysis 38(6), 2047–2078 (2001)
  4. Journal of Computational Physics 356, 410–438 (2018)
  5. Journal of Computational Physics 392, 161–186 (2019)
  6. Journal of Scientific Computing 75(1), 83–110 (2018)
  7. Dubiner, M.: Spectral methods on triangles and other domains. Journal of Scientific Computing 6, 345–390 (1991)
  8. Felippa, C.A.: A compendium of FEM integration formulas for symbolic work. Engineering Computations 21(8), 867–890 (2004)
  9. Journal of Computational Physics 252, 518–557 (2013)
  10. Journal of Scientific Computing 77(1), 154–200 (2018)
  11. Journal of Engineering Mathematics 56, 307–322 (2006)
  12. Hicken, J.E.: SummationByParts.jl. Version 0.1.2 (2015). URL https://github.com/OptimalDesignLab/SummationByParts.jl
  13. SIAM Journal on Scientific Computing 38(4), A1935–A1958 (2016)
  14. Journal of Computational and Applied Mathematics 237(1), 111–125 (2013)
  15. Journal of Computational Physics 228(15), 5410–5436 (2009)
  16. In: Proceedings of ICNN’95 - International Conference on Neural Networks, vol. 4, pp. 1942–1948. IEEE (1995)
  17. Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: R.A. Askey (ed.) Theory and Application of Special Functions, pp. 435–495. Academic Press (1975)
  18. Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Quarterly of Applied Mathematics 2(2), 164–168 (1944)
  19. Journal of Computational Physics 336, 458–480 (2017)
  20. Journal of Computational Physics 411, 109380 (2020)
  21. Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics 11(2), 431–441 (1963)
  22. arXiv preprint arXiv:2306.05975 (2023)
  23. Mulder, W.A.: Higher-order mass-lumped finite elements for the wave equation. Journal of Computational Acoustics 9(02), 671–680 (2001)
  24. J. Comput. Physics 290, 132–138 (2015)
  25. In: Proceedings of the Practice and Experience in Advanced Research Computing on Rise of the Machines (learning), pp. 1–8 (2019)
  26. Proriol, J.: Sur une famille de polynomes à deux variables orthogonaux dans un triangle. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 245(26), 2459–2461 (1957)
  27. Ranocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler equations. Journal of Scientific Computing 76(1), 216–242 (2018)
  28. Journal of Scientific Computing 82(3), 70 (2020)
  29. SIAM Journal on Numerical Analysis 38(5), 1707–1720 (2000)
  30. In: 21st AIAA Computational Fluid Dynamics Conference, AIAA 2013-2830 (2013)
  31. Applied Numerical Mathematics 58(7), 1030–1041 (2008)
  32. Computers & Mathematics with Applications 69(10), 1232–1241 (2015)
  33. Journal of Scientific Computing 61, 398–423 (2014)
  34. Journal of Computational Physics 445, 110634 (2021)
  35. arXiv preprint arXiv:2305.07181 (2023)
  36. Computers & Mathematics with Applications 82, 179–199 (2021)
  37. Journal of Scientific Computing 75(3), 1385–1414 (2018)
  38. Journal of Computational Mathematics pp. 89–96 (2009)
Citations (3)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now