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Journal of Scientific Computing

Erschienen in:

17.08.2017

A Projection Method for the Conservative Discretizations of Parabolic Partial Differential Equations

verfasst von: Darae Jeong, Junseok Kim

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2018

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Abstract

We present a projection method for the conservative discretizations of parabolic partial differential equations. When we solve a system of discrete equations arising from the finite difference discretization of the PDE, we can use iterative algorithms such as conjugate gradient, generalized minimum residual, and multigrid methods. An iterative method is a numerical approach that generates a sequence of improved approximate solutions for a system of equations. We repeat the iterative algorithm until a numerical solution is within a specified tolerance. Therefore, even though the discretization is conservative, the actual numerical solution obtained from an iterative method is not conservative. We propose a simple projection method which projects the non-conservative numerical solution into a conservative one by using the original scheme. Numerical experiments demonstrate the proposed scheme does not degrade the accuracy of the original numerical scheme and it preserves the conservative quantity within rounding errors.

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Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)CrossRef

Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)CrossRef

Lee, D., Huh, J.Y., Jeong, D., Shin, J., Yun, A., Kim, J.: Physical, mathematical, and numerical derivations of the Cahn–Hilliard equation. Comp. Mater. Sci. 81, 216–225 (2014)CrossRef

Li, X., Zhang, L., Wang, S.: A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator. Appl. Math. Comput. 219, 3187–3197 (2012)MathSciNetMATH

Fei, Z., Perez-Garcia, V.M., Vazquez, L.: Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Appl. Math. Comput. 71, 165–177 (1995)MathSciNetMATH

Chang, Q., Xu, L.: A numerical method for a system of generalized nonlinear Schrödinger equations. J. Comput. Math. 4, 191–199 (1986)MathSciNetMATH

Chang, Q., Wang, G.: Multigrid and adaptive algorithm for solving the nonlinear Schrödinger equation. J. Comput. Phys. 88, 362–380 (1990)MathSciNetCrossRefMATH

Chang, Q., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 148, 397–415 (1999)MathSciNetCrossRefMATH

Matsuo, T., Furihata, D.: Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171(2), 425–447 (2001)MathSciNetCrossRefMATH

10.

Chang, Q., Wang, G., Guo, B.: Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary notion. J. Comput. Phys. 93, 360–375 (1991)MathSciNetCrossRefMATH

11.

Zhang, F., Vazquez, L.: Two energy conserving numerical schemes for the Sine–Gordon equation. Appl. Math. Comput. 45, 17–30 (1991)MathSciNetCrossRefMATH

12.

Wong, Y.S., Chang, Q., Gong, L.: An initial-boundary value problem of a nonlinear Klein–Gordon equation. Appl. Math. Comput. 84, 77–93 (1997)MathSciNetMATH

13.

Chang, Q., Jiang, H.: A conservative scheme for the Zakharov equation. J. Comput. Phys. 113, 309–319 (1994)MathSciNetCrossRefMATH

14.

Chang, Q., Guo, B., Jiang, H.: Finite difference method for generalized Zakharov equation. J. Comput. Phys. 113, 309–319 (1994)MathSciNetCrossRef

15.

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

16.

Choo, S.M., Chung, S.K.: Conservative nonlinear difference scheme for the Cahn–Hilliard equation. Comput. Math. Appl. 36, 31–39 (1998)MathSciNetCrossRefMATH

17.

Furihata, D., Matsuo, T.: A stable, convergent, conservative and linear finite difference scheme for the Cahn–Hilliard equation. Jpn. J. Ind. Appl. Math. 20, 65–85 (2003)MathSciNetCrossRefMATH

18.

De Mello, E.V.L., da Silveira Filho, O.T.: Numerical study of the Cahn–Hilliard equation in one, two and three dimensions. Phys. A. 347, 429–443 (2005)MathSciNetCrossRef

19.

Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193, 511–543 (2004)MathSciNetCrossRefMATH

20.

Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Brooks Cole, Boston (2011)MATH

21.

Hackbusch, W.: Multi-grid Methods and Applications. Springer, Berlin (1985)CrossRefMATH

22.

Briggs, W.L., McCormick, S.F.: A Multigrid Tutorial. SIAM, Philadelphia (2000)CrossRefMATH

23.

Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic press, London (2000)MATH

24.

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

25.

Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856–869 (1986)MathSciNetCrossRefMATH

26.

Elliott, C.M.: The Cahn–Hilliard Model for the Kinetics of Phase Separation Number 88 in International Series of Numerical Mathematics. Birkhauser, Basel (1989)

27.

Cahn, J.W., Chow, S.N., van Vleck, E.S.: Spatially iscrete nonlinear diffusion equations. Rocky Mt. J. Math. 25(1), 87–118 (1995)CrossRefMATH

28.

Maier, R.S., Rath, W., Petzold, L.R.: Parallel solution of large-scale differential-algebraic systems. Concurr. Comput.-Pract. E. 7(8), 795–822 (1995)

29.

Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. In: MRS Proceedings, 529 pp. 39. Cambridge University Press (1998)

30.

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

31.

Guillet, T., Teyssier, R.: A simple multigrid scheme for solving the Poisson equation with arbitrary domain boundaries. J. Comput. Phys. 230(12), 4756–4771 (2011)MathSciNetCrossRefMATH

32.

Fulton, S.R., Ciesielski, P.E., Schubert, W.H.: Multigrid methods for elliptic problems: a review. Mon. Weather Rev. 114(5), 943–959 (1986)CrossRef

33.

Lee, C., Jeong, D., Shin, J., Li, Y., Kim, J.: A fourth-order spatial accurate and practically stable compact scheme for the Cahn–Hilliard equation. Phys. A. 409(1), 17–28 (2014)MathSciNetCrossRef

34.

Kim, J.: Three-dimensional numerical simulations of a phase-field model for anisotropic interfacial energy. Commun. Korean. Math. Soc. 22(3), 453–464 (2007)MathSciNetCrossRefMATH

35.

Kim, J.: A diffuse-interface model for axisymmetric immiscible two-phase flow. Appl. Math. Comput. 160(2), 589–606 (2005)MathSciNetMATH

36.

Briggs, W.: A Multigrid Tutorial. SIAM, Philadelphia (1977)MATH

37.

Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic press, London (2000)MATH

38.

Shin, J., Jeong, D., Kim, J.: A conservative numerical method for the Cahn–Hilliard equation in complex domains. J. Comput. Phys. 230(19), 7441–7455 (2011)MathSciNetCrossRefMATH

Titel: A Projection Method for the Conservative Discretizations of Parabolic Partial Differential Equations
verfasst von: Darae Jeong
Junseok Kim
Publikationsdatum: 17.08.2017
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2018
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0536-2

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