52
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A parallel high-order accuracy algorithm for the Helmholtz equations

&
Pages 56-94 | Received 05 May 2023, Accepted 02 Dec 2023, Published online: 08 Feb 2024

References

  • I.M. Babuška and S.A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42(3) (2000), pp. 451–484.
  • R.I. Balam and M.U. Zapata, A new eighth-order implicit finite difference method to solve the three-dimensional Helmholtz equation, Comput. Math. Appl. 80(5) (2020), pp. 1176–1200.
  • A. Bayliss, C.I. Goldstein, and E. Turkel, An iterative method for Helmholtz equation, J. Comput. Phys. 49(3) (1983), pp. 443–457.
  • J. Biazar and R. Asayesh, An efficient high-order compact finite difference method for the Helmholtz equation, Comput. Methods Differ. Equations 8(3) (2020), pp. 553–563.
  • C. Chalons, M.L. Delle Monache, and P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem, Interfaces Free. Boundaries 19(4) (2017), pp. 553–570.
  • K. Chen and P.J. Harris, Efficient preconditioners for iteration solution of the boundary element equations for the three-dimensional Helmholtz equation, Appl. Numer. Math. 36(4) (2001), pp. 475–489.
  • D. Cheng, Z. Liu, and T. Wu, A multigrid-based preconditioned solver for the Helmholtz equation with a discretization by 25-point difference scheme, Math. Comput. Simul. 117 (2015), pp. 54–67.
  • D. Cheng, Z. Liu, G. Xue, and Y. Gao, High-performance parallel preconditional iterative solver for the Helmholtz equation with large wavenumbers, Comput. Sci. 45(7) (2018), pp. 299–306.
  • P.C. Chu and C. Fan, A three-point combined compact difference scheme, J. Comput. Phys. 140 (1998), pp. 370–399.
  • M.P. D'Arienzo and L. Rarità, Management of supply chains for the wine production, in International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2019, 23 September 2019 – 28 September 2019, AIP Conf Proc. 2293, Rhodes, Greece, Vol. 24, 2020, p. 420042.
  • M.P. D'Arienzo and L. Rarità, Growth effects on the network dynamics with applications to the cardiovascular system, in International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2019, 23 September 2019 – 28 September 2019, AIP Conf Proc., Rhodes, Greece, Vol. 2293, 2020, p. 420041.
  • M. de Falco, M. Gaeta, V. Loia, L. Rarità, and S. Tomasiello, Differential quadrature-based numerical solutions of a fluid dynamic model for supply chains, Commun. Math. Sci. 14 (2016), pp. 1467–1476.
  • M.L. Delle Monache and P. Goatin, A numerical scheme for moving bottlenecks in traffic flow, B. Braz. Math. Soc. 47(2) (2016), pp. 605–617.
  • X. Feng, A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient, Int. J. Comput. Math. 89(5) (2012), pp. 618–624.
  • X. Feng, Z. Li, and Z. Qiao, High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients, J. Comput. Math. 29(3) (2011), pp. 324–340.
  • M. Fournié and S. Karaa, Iterative methods and high-order difference schemes for 2D elliptic problems with mixed derivative, J. Appl. Math. Comput. 22(3) (2006), pp. 349–363.
  • Y. Fu, Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers, J. Comput. Math. 26(1) (2008), pp. 98–111.
  • D. Gordon and R. Gordon, Parallel solution of high frequency Helmholtz equations using high order finite difference schemes, Appl. Math. Comput. 218(21) (2012), pp. 10737–10754.
  • D. Gordon and R. Gordon, Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers, J. Comput. Appl. Math. 237(1) (2013), pp. 182–196.
  • Y. Gryazin, Preconditioned Krylov subspace methods for sixth order compact approximations of the Helmholtz equation, ISRN Comput. Math. 2014 (2014), p. 745849.
  • A. Handlovičovà and I. Riečanovà, Numerical solution to the complex 2D Helmholtz equation based on finite volume method with impedance boundary conditions, Open Phys. 14(1) (2016), pp. 436–443.
  • F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number part I: The h-version of the FEM, Comput. Math. Appl. 30(9) (1995), pp. 9–37.
  • S.J. Kahlaf and A.A. Mhassin, Numerical solution of a two-dimensional Helmholtz equation with Dirichlet boundary conditions, J. Interdiscip. Math. 24(4) (2021), pp. 971–982.
  • H. Knibbe, C.W. Oosterlee, and C. Vuik, GPU implementation of a Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method, J. Comput. Appl. Math. 236(3) (2011), pp. 281–293.
  • N. Kumar and R.K. Dubey, A new development of sixth order accurate compact scheme for the Helmholtz equation, J. Appl. Math. Comput. 62(1–2) (2020), pp. 637–662.
  • L. Liao and G. Zhang, Preconditioning of complex linear systems from the Helmholtz equation, Comput. Math. Appl. 72(9) (2016), pp. 2473–2485.
  • T. Ma and Y. Ge, A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation, Adv. Differ. Equations. 2019 (2019), p. 98.
  • J.E. Macías-Díaz, A. Raza, N. Ahmed, and M. Rafiq, Analysis of a nonstandard computer method to simulate a nonlinear stochastic epidemiological model of coronavirus-like diseases, Comput. Meth. Prog. Bio. 204 (2021), p. 106054.
  • R.P. Manohar and J.W. Stephenson, Single cell high order difference methods for the Helmholtz equation, J. Comput. Phys. 51(3) (1983), pp. 444–453.
  • M. Nabavi, M.H.K. Siddiqui, and J. Dargahi, A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation, J. Sound Vib. 307(3–5) (2007), pp. 972–982.
  • G. Ortega, J. Lobera, I. Garcia, A.M. Pilar, and E.M. Garzon, Parallel resolution of the 3D Helmholtz equation based on multi-graphics processing unit clusters, Concurr. Comput. Pract. Exp. 27(13) (2015), pp. 3205–3219.
  • Ö. Oruç, A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations, Comput. Math. Appl. 77 (2019), pp. 1799–1820.
  • Ö. Oruç, A meshfree computational approach based on multiple-Scale pascal polynomials for numerical solution of a 2D elliptic problem with nonlocal boundary conditions, Int. J. Comput. Methods. 10 (2020), p. 17.
  • Ö. Oruç, An efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solution of 2-D and 3-D second order elliptic interface problems, J. Comput. Phys. 428 (2021), p. 110070.
  • C.D. Riyanti, A. Kononov, Y.A. Erlangga, C. Vuik, C.W. Oosterlee, R.E. Plessix, and W.A. Mulder, A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation, J. Comput. Phys. 224(1) (2007), pp. 431–448.
  • I. Singer and E. Turkel, High-order finite difference methods for the Helmholtz equation, Comput. Methods Appl. Mech. Eng. 163(1–4) (1998), pp. 343–358.
  • X.H. Sun and Y. Zhuang, A high-order direct solver for Helmholtz equations with Neumann boundary conditions, NASA ICASE Technical Report No. 97-11. NASA Langley Research Center, Hampton, VA, 1997.
  • G. Sutmann, Compact finite difference schemes of sixth order for the Helmholtz equation, J. Comput. Appl. Math. 203(1) (2007), pp. 15–31.
  • E. Turkel, D. Gordon, R. Gordon, and S. Tsynkov, Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number, J. Comput. Phys. 232(1) (2013), pp. 272–287.
  • Z. Wang, Y.B. Ge, H.W. Sun, and T. Sun, Sixth-order quasi-compact difference schemes for 2D and 3D Helmholtz equations, Appl. Math. Comput. 431 (2022), p. 127347.
  • H. Wu, FEM and CPI-FEM for Helmholtz equation with high wave number, Math. Numer. Sin. 40(2) (2018), pp. 191–213.
  • T. Wu, Y. Sun, and D. Cheng, A new finite difference scheme for the 3D Helmholtz equation with a preconditioned iterative solver, Appl. Numer. Math. 161 (2021), pp. 348–371.
  • T. Wu and R. Xu, An optimal compact sixth-order finite difference scheme for the Helmholtz equation, Comput. Math. Appl. 75(7) (2018), pp. 2520–2537.
  • R. Xu and T. Wu, Finite volume method for solving the stochastic Helmholtz equation, Adv. Differ. Equations 2019 (2019), p. 84.
  • L. Zhang, X. Gong, J. Song, and J. Hu, Parallel preconditioned GMRES solvers for 3-D Helmholtz equations in regional non-hydrostatic atmosphere model, in International Conference on Computer Science & Software Engineering, IEEE, Wuhan, China, 2008, pp. 287–290.
  • J. Zhang, G. Gu, J. Shi, and M. Zhao, High-order fast algorithm for solving Helmholtz equation with large wave numbers, in 2nd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC), ELECTR NETWORK, Kunming, China, 2022, pp. 25–27.
  • Y. Zhang, K. Wang, and R. Guo, Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition, Adv. Differ. Equations 2019 (2019), p. 362.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.