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Journal of Scientific Computing

Erschienen in:

01.01.2013

A Family of Fourth-Order and Sixth-Order Compact Difference Schemes for the Three-Dimensional Poisson Equation

verfasst von: Shuying Zhai, Xinlong Feng, Yinnian He

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2013

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Abstract

In this paper a family of fourth-order and sixth-order compact difference schemes for the three dimensional (3D) linear Poisson equation are derived in detail. By using finite volume (FV) method for derivation, the highest-order compact schemes based on two different types of dual partitions are obtained. Moreover, a new fourth-order compact scheme is gained and numerical experiments show the new scheme is much better than other known fourth-order schemes. The outline for the nonlinear problems are also given. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference scheme.

Vorheriger Artikel A Novel Hierarchial Error Estimate for Elliptic Obstacle Problems

Nächster Artikel Numerical Study of the Plasma-Lorentz Model in Metamaterials

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Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979) CrossRef

Ananthakrishnaiah, U., Monahar, R., Stephenson, J.W.: Fourth-order finite difference methods for three-dimensional general linear elliptic problems with variable coefficients. Numer. Methods Partial Differ. Equ. 3, 229–240 (1987) MATHCrossRef

Balakrishnan, K., Ramachandran, P.A.: Osculatory interpolation in the method of fundamental solution for nonlinear Poisson problems. J. Comput. Phys. 172, 1–18 (2001) MathSciNetMATHCrossRef

Bouchon, F., Peichl, G.H.: A second-order immersed interface technique for an elliptic Neumann problem. Numer. Methods Partial Differ. Equ. 23, 400–420 (2007) MathSciNetMATHCrossRef

Dumett, M.A., Keener, J.P.: An immersed interface method for anisotropic elliptic problems on irregular domains in 2D. Numer. Methods Partial Differ. Equ. 21(2), 397–420 (2005) MathSciNetMATHCrossRef

Dumett, M.A., Keener, J.P.: An immersed interface method for solving anisotropic elliptic boundary value problems in three dimensions. SIAM J. Sci. Comput. 25(1), 348–367 (2003) MathSciNetMATHCrossRef

Furihata, D.: A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87, 675–699 (2001) MathSciNetMATHCrossRef

Feng, X.L., He, Y.N.: High order iterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186(2), 1617–1623 (2007) MathSciNetMATHCrossRef

Feng, X.L., Li, R.F., He, Y.N., Liu, D.M.: P ₁-Nonconforming quadrilateral finite volume methods for the semilinear elliptic equations. J. Sci. Comput. doi:10.1007/s10915-011-9557-4

10.

Feng, X.B., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003) MathSciNetMATHCrossRef

11.

Ge, L.X., Zhang, J.: Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients. J. Comput. Appl. Math. 143(1), 9–27 (2002) MathSciNetMATHCrossRef

12.

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

13.

Ge, Y.B., Tian, Z.F., Ma, H.L.: A high accuracy multigrid method for the three-dimensional Poission equation. Appl. Math. 19(2), 313–318 (2006) MathSciNetMATH

14.

Jomma, Z., Macaskill, C.: The Shortley-Weller embedded finite-difference method for the 3D Poission equation with mixed boundary conditions. J. Comput. Phys. 229, 3675–3690 (2010) MathSciNetCrossRef

15.

Kwon, Y., Stephenson, J.W.: Single cell finite difference approximations for Poisson’s equation in three variables. Appl. Math. Notes 2, 13 (1982) MathSciNet

16.

Kyei, Y., Roop, J.P., Tang, G.Q.: A family of sixth-order compact finite-difference schemes for the three-dimensional Poisson equation. Adv. Numer. Anal. 1–18 (2010)

17.

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

18.

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, 1137–1151 (1995) MathSciNetMATHCrossRef

19.

Li, M., Tang, T.: A compact fourth-order finite difference scheme for unsteady viscous incompressible flows. J. Sci. Comput. 16, 29–46 (2001) MathSciNetMATHCrossRef

20.

Li, Q.Y., Wang, N.C., Ri, D.Y.: Numerical Analysis. Tsinghua University Press, Beijing (2001) (in Chinese)

21.

LeVeque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994) MathSciNetMATHCrossRef

22.

Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992) MathSciNetMATHCrossRef

23.

Liu, X., Fedkiw, R., Kang, M.: A boundary condition capturing method for Poisson’s equation on irregular domains. J. Comput. Phys. 160, 151–178 (2000) MathSciNetMATHCrossRef

24.

Manohar, R., Stephenson, J.W.: High order difference schemes for linear partial differential equations. SIAM J. Sci. Stat. Comput. 5(1), 69–77 (1984) MathSciNetMATHCrossRef

25.

Ma, Y.Z., Ge, Y.B.: A high order finite difference method with Richardson extrapolation for 3D convection diffusion equation. Appl. Math. Comput. 215, 3408–3417 (2010) MathSciNetMATHCrossRef

26.

Peichl, G.H.: An immersed interface technique for mixed boundary value problems. In: Proceedings of the SEAMS-GMU Conference, Yogyakarta, pp. 14–23 (2003)

27.

Ramière, I., Angot, P., Belliard, M.: A fictitious domain approach with spread interface for elliptic problems with general boundary conditions. Comput. Methods Appl. Mech. Eng. 196, 766–781 (2007) MATHCrossRef

28.

Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. Wadsworth & Brooks-Cole Advanced Books & Software, Pacific Grove (1989) MATH

29.

Sakurai, K., Aoki, T., Lee, W.H., Kato, K.: Poisson equations solver with fourth-order accuracy by using interpolated differential operator scheme. Comput. Math. Appl. 43, 621–630 (2002) MathSciNetMATHCrossRef

30.

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

31.

Spotz, W.F.: High-order compact finite difference schemes for computational mechanics. Ph.D. Thesis, University of Texas at Austin, Austin, TX (1995)

32.

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

33.

Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, San Diego (2001) MATH

34.

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

35.

Wang, K., Feng, X.L.: New predictor-corrector methods of second-order for solving nonlinear equations. Int. J. Comput. Math. 88(2), 296–313 (2011) MathSciNetMATH

36.

Xu, C.F.: A new numerical method for solving dinite element equations-Iteration in subspace in successive levels. J. Huazhong Univ. of Scf. & Tech. 105–112 (1985)

37.

Xu, C.F.: Numerical Solution of Practical Partial Differential Equations. Huazhong University of Science & Technology Press, Hubei (2003)

38.

Zhai, S.Y., Feng, X.L., He, Y.N.: A new method to deduce high-order compact difference schemes for the two dimensional elliptic equations. Preprint

39.

Zhai, S.Y., Feng, X.L., Liu, D.M.: A new family of high-order compact difference schemes for the three dimensional convection-diffusion equation with variable coefficients. Preprint

Titel: A Family of Fourth-Order and Sixth-Order Compact Difference Schemes for the Three-Dimensional Poisson Equation
verfasst von: Shuying Zhai
Xinlong Feng
Yinnian He
Publikationsdatum: 01.01.2013
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2013
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-012-9607-6

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