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Sixth-order compact difference scheme and multigrid method for solving the Poisson equation

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Abstract

This paper proposes a sixth-order compact difference scheme of Poisson equation based on the sixth-order compact difference operator of the second derivative. The biggest difference between the proposed scheme and other sixth-order scheme is that the right hand contains second partial derivation of source term; this term makes the proposed scheme more accurate than other sixth-order schemes. The proposed scheme is combined with the multigrid method to solve two- and three-dimensional Poisson equations with Dirichlet boundary conditions. The result is compared with other sixth-order schemes in several numerical experiments. The numerical results show that the proposed scheme achieves the desired accuracy and has smaller errors than other schemes of the same order. Further, the multigrid method is higher efficient than traditional iterative method in accelerating the convergence.

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References

  1. Ames, W.F.: Numerical Methods for Partial Differential Equations. Academic Press, New York (1977)

    Google Scholar 

  2. Lu, J.P., Guan, Z.: Numerical Solution of Partial Differential Equations. Qinghua University Press, Beijing (1987)

    Google Scholar 

  3. Hirsh, R.: Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. Comput. Phys. 19(1), 90–109 (1975)

    Article  MathSciNet  Google Scholar 

  4. Tolstykh, A.: High Accuracy Non-centered Compact Difference Schemes for Fluid Dynamics Applications. World Scientific, New Jersey (1994)

    Book  Google Scholar 

  5. Tian, Z.F.: An high-order compact finite-difference schemes for the second dimensional Poisson equation. J. Northwest Univ. 26(2), 109–114 (1996)

    MathSciNet  Google Scholar 

  6. Ciment, M., Leventhal, S.: Higher order compact implicit schemes for the wave equation. Math. Comput. 132(29), 985–994 (1975)

    Article  MathSciNet  Google Scholar 

  7. Zhang, J.: Multigrid method and fourth order compact difference scheme for 2D Poisson equation with unequal meshsize discretization. J. Comput. Phys. 179(1), 170–179 (2002)

    Article  Google Scholar 

  8. Wang, Y., Zhang, J.: Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation. J. Comput. Phys. 228(1), 137–146 (2009)

    Article  MathSciNet  Google Scholar 

  9. Mohanty, R.K., Singh, S.: A new fourth order discretization for singularly perturbed two-dimensional non-linear elliptic boundary value problems. Appl. Math. Comput. 175(2), 1400–1414 (2006)

    MathSciNet  Google Scholar 

  10. Zhong, W.J., Wang, J., Zhang, J.: A general meshsize fourth-order compact difference discretization scheme for 3D Poission equation. Appl. Math. Comput. 183(2), 804–812 (2006)

    MathSciNet  Google Scholar 

  11. Gupta, M.M., Kouatchou, J., Zhang, J.: Comparison of second-order and fourth-order discretization for multigrid Poisson solvers. J. Comput. Phys. 132(2), 226–232 (1997)

    Article  MathSciNet  Google Scholar 

  12. Cao, F.J., Ge, Y.B., Zhang, J.: A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on nonuniform grids. J. Comput. Phys. 234(1), 199–216 (2013)

    MathSciNet  Google Scholar 

  13. Wang, H.Q., Zhang, Y., Ma, X., et al.: An efficient implementation of fourth-order compact finite difference scheme for Poisson equation with Dirichlet boundary conditions. Comput. Math. Appl. 71(9), 1843–1860 (2016)

    Article  MathSciNet  Google Scholar 

  14. Zapata, M.U., Balam, R.I.: High-order implicit finite difference schemes for the two-dimensional Poisson equation. Appl. Math. Comput. 309(15), 222–244 (2017)

    MathSciNet  Google Scholar 

  15. Li, M., Fornberg, B., Tang, T.: A compact fourth order finite difference scheme for the steady incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids. 20(10), 1137–1151 (1995)

    Article  MathSciNet  Google Scholar 

  16. Zhang, J.: Fast and high accuracy multigrid solution of the three-dimensional Poisson equation. J. Comput. Phys. 143(2), 161–449 (1998)

    Article  MathSciNet  Google Scholar 

  17. Spotz, W.F., Carey, G.F.: High-order compact scheme for the steady stream-function vorticity equations. Int. J. Numer. Methods Eng. 38(20), 3497–3512 (1995)

    Article  MathSciNet  Google Scholar 

  18. Spotz, W.F., Carey, G.F.: A high-order compact formulation for the 3D Poisson equation. Numer. Methods Partial Diff. Equ. 12(2), 235–243 (1996)

    Article  Google Scholar 

  19. Zhai, S.Y., Feng, X.L., He, Y.N.: A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation. Appl. Math. Comput. 230(1), 9–26 (2014)

    MathSciNet  Google Scholar 

  20. Gatiso, A.H., Belachew, M.T., Wolle, G.A.: Sixth-order compact finite difference scheme with discrete sine transform for solving Poisson equations with Dirichlet boundary conditions. Results Appl. Math. 10, 100148 (2021)

    Article  MathSciNet  Google Scholar 

  21. Zapata, M.U., Balam, R.I., Montalvo-Urquizo, J.: A compact sixth-order implicit immersed interface method to solve 2D Poisson equations with discontinuities. Math. Comput. Simul. 210, 384–407 (2023)

    Article  MathSciNet  Google Scholar 

  22. Hu, H.L., Li, M., Pan, K.J., Wu, P.X.: An extrapolation accelerated multiscale Newton-MG method for fourth-order compact discretizations of semilinear Poisson equations. Comput. Math. Appl. 113, 189–197 (2022)

    Article  MathSciNet  Google Scholar 

  23. Deriaz, E.: High-order adaptive mesh refinement multigrid Poisson solver in any dimension. J. Comput. Phys. 480, 112012 (2023)

    Article  MathSciNet  Google Scholar 

  24. Hu, S.G., Pan, K.J., Wu, X.X., Ge, Y.B., Li, Z.L.: An efficient extrapolation multigrid method based on a HOC scheme on nonuniform rectilinear grids for solving 3D anisotropic convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 403, 115724 (2023)

    Article  MathSciNet  Google Scholar 

  25. Nikan, O., Avazzadeh, Z.: Coupling of the Crank-Nicolson scheme and localized meshless technique for viscoelastic wave model in fluid flow. J. Comput. Appl. Math. 398, 113695 (2021)

    Article  MathSciNet  Google Scholar 

  26. Nikan, O., Avazzadeh, Z., Rasoulizadeh, M.N.: Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory. Nonlinear Dyn. 106, 783–813 (2021)

    Article  Google Scholar 

  27. Nikan, O., Avazzadeh, Z.: A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics. Appl. Math. Comput. 401, 126063 (2021)

    MathSciNet  Google Scholar 

  28. Sutmann, G., Steffen, B.: High-order compact solvers for the three-dimensional Poisson equation. J. Comput. Appl. Math. 187(2), 142–170 (2006)

    Article  MathSciNet  Google Scholar 

  29. Ge, Y.B., Cao, F.J.: Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems. J. Comput. Phys. 230(10), 4051–4070 (2011)

    Article  MathSciNet  Google Scholar 

  30. Wesseling, P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992)

    Google Scholar 

  31. Ge, Y.B.: Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D Poisson equation. J. Comput. Phys. 229(18), 6381–6391 (2010)

    Article  MathSciNet  Google Scholar 

  32. Gupta, M.M., Zhang, J.: High accuracy multigrid solution of the 3D convection-diffusion equation. Appl. Math. Comput. 113(2–3), 249–274 (2000)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to the anonymous referees for their useful suggestions and comments that improved the presentation of this paper.

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Authors and Affiliations

  1. School of Civil and Hydraulic Engineering, Ningxia University, Yinchuan, 750021, China

    Xiaogang Li

  2. School of Science, Dalian Minzu University, Dalian, 116600, China

    Yongbin Ge

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  1. Xiaogang Li
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  2. Yongbin Ge
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Correspondence to Yongbin Ge.

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This work is partially supported by National Natural Science Foundation of China (12161067), National Natural Science Foundation of Ningxia (2022AAC02023,2022AAC03070), National Youth Top-notch Talent Support Program of Ningxia.

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Li, X., Ge, Y. Sixth-order compact difference scheme and multigrid method for solving the Poisson equation. Math Sci (2024). https://doi.org/10.1007/s40096-023-00522-3

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  • DOI: https://doi.org/10.1007/s40096-023-00522-3

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