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Journal of Scientific Computing

Erschienen in:

01.01.2021

Stability and Error Estimate of the Operator Splitting Method for the Phase Field Crystal Equation

verfasst von: Shuying Zhai, Zhifeng Weng, Xinlong Feng, Yinnian He

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2021

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Abstract

In this paper, we propose a second-order fast explicit operator splitting method for the phase field crystal equation. The basic idea lied in our method is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using the Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size. The nonlinear one is solved via second-order strong stability preserving Runge–Kutta method. The stability and convergence are discussed in \(L^2\)-norm. Numerical experiments are performed to validate the accuracy and efficiency of the proposed method. Moreover, energy degradation and mass conservation are also verified.

Vorheriger Artikel FFT-Based High Order Central Difference Schemes for Poisson’s Equation with Staggered Boundaries

Nächster Artikel Two New Variants of the Simpler Block GMRES Method with Vector Deflation and Eigenvalue Deflation for Multiple Linear Systems

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Elder, K.R., Katakowski, M., Haataja, M., Grant, M.: Modeling elasticity in crystal growth. Phys. Rev. Lett. 88(24), 245701 (2002)CrossRef

Elder, K.R., Grant, M.: Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 70(5), 051605 (2004)CrossRef

Berry, J., Provatas, N., Rottler, J., Sinclair, C.W.: Defect stability in phase-field crystal models: stacking faults and partial dislocations. Phys. Rev. B 86, 224112 (2012)CrossRef

Provatas, N., Dantzig, J., Athreya, B., Chan, P., Stefanovic, P., Goldenfeld, N., Elder, K.: Using the phase-field crystal method in the multi-scale modeling of microstructure evolution. JOM 59, 83–90 (2007)CrossRef

Stolle, J., Provatas, N.: Characterizing solute segregation and grain boundary energy in binary alloy phase field crystal models. Comput. Mater. Sci. 81, 493–502 (2014)CrossRef

Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319 (1977)CrossRef

Wang, C., Wise, S.M., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47, 2269–2288 (2009)MathSciNetMATHCrossRef

Wang, C., Wise, S.M.: An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 49, 945–969 (2011)MathSciNetMATHCrossRef

Baskaran, A., Hu, Z.Z., Lowengrub, J.S., Wang, C., Wise, S.M., Zhou, P.: Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation. J. Comput. Phys. 250, 270–292 (2013)MathSciNetMATHCrossRef

10.

Baskaran, A., Lowengrub, J.S., Wang, C., Wise, S.M.: Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 51, 2851–2873 (2013)MathSciNetMATHCrossRef

11.

Li, Q., Mei, L.Q., You, B.: A second-order, uniquely solvable, energy stable BDF numerical scheme for the phase field crystal model. Appl. Numer. Math. 134, 46–65 (2018)MathSciNetMATHCrossRef

12.

Xia, B.H., Mei, C.L., Yu, Q., Li, Y.B.: A second order unconditionally stable scheme for the modified phase field crystal model with elastic interaction and stochastic noise effect. Comput. Methods Appl. Mech. Eng. 363, 112795 (2020)MathSciNetMATHCrossRef

13.

Cheng, K.L., Wang, C., Wise, S.M.: An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation. arXiv:1906.12255 [math.NA]

14.

Yang, X.F., Han, D.Z.: Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model. J. Comput. Phys. 330, 1116–1134 (2017)MathSciNetMATHCrossRef

15.

Zhao, J., Wang, Q., Yang, X.F.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int. J. Numer. Methods Eng. 110, 279–300 (2017)MathSciNetMATHCrossRef

16.

Li, Q., Mei, L.Q., Yang, X.F., Li, Y.B.: Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation. Adv. Comput. Math. 45, 1551–1580 (2019)MathSciNetMATHCrossRef

17.

Liu, Z.G., Li, X.L.: Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation. Appl. Numer. Math. 150, 491–506 (2020)MathSciNetMATHCrossRef

18.

Liu, Z.G., Li, X.L.: Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation. Numer. Algorithms 85, 107–132 (2020)MathSciNetMATHCrossRef

19.

Li, X.L., Shen, J.: Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation. Adv. Comput. Math. 46, 48 (2020)MathSciNetMATHCrossRef

20.

Li, X.L., Shen, J.: Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation. arXiv:2004.04319 [math.NA]

21.

Goldman, D., Kaper, T.: \(N\)th-order operator splitting schemes and nonreversible systems. SIAM J. Numer. Anal. 33, 349–367 (1996)MathSciNetMATHCrossRef

22.

Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)MathSciNetMATHCrossRef

23.

Yanenko, N.N.: The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables. Springer, New York (1971)MATHCrossRef

24.

Holden, H., Karlsen, K.H., Risebro, N.H.: Operator splitting methods for generalized Korteweg-de Vries equations. J. Comput. Phys. 153, 203–222 (1999)MathSciNetMATHCrossRef

25.

Cheng, Y.Z., Kurganov, A., Qu, Z.L., Tang, T.: Fast and stable explicit operator splitting methods for phase-field models. J. Comput. Phys. 303, 45–65 (2015)MathSciNetMATHCrossRef

26.

Zhai, S.Y., Weng, Z.F., Feng, X.L.: Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen–Cahn model. Appl. Math. Model. 40, 1315–1324 (2016)MathSciNetMATHCrossRef

27.

Zhai, S.Y., Wu, L.Y., Wang, J.Y., Weng, Z.F.: Numerical approximation of the fractional Cahn–Hilliard equation by operator splitting method. Numer. Algorithms 84, 1155–1178 (2020)MathSciNetMATHCrossRef

28.

Li, X., Qiao, Z.H., Zhang, H.: Convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection. SIAM J. Numer. Anal. 55, 265–285 (2017)MathSciNetMATHCrossRef

29.

Bao, W.Z., Li, H.L., Shen, J.: A generalized-Laguerre–Fourier–Hermite pseudospectral method for computing the dynamics of rotating Bose–Einstein condensates. SIAM J. Sci. Comput. 31, 3685–3711 (2009)MathSciNetMATHCrossRef

30.

Shen, J., Wang, Z.Q.: Error analysis of the Strang time-splitting Laguerre–Hermite/Hermite collocation methods for the Gross–Pitaevskii equation. Found. Comput. Math. 13, 99–137 (2013)MathSciNetMATHCrossRef

31.

Zhang, C., Huang, J.F., Wang, C., Yue, X.Y.: On the operator splitting and integral equation preconditioned deferred correction methods for the “Good” Boussinesq equation. J. Sci. Comput. 75, 687–712 (2018)MathSciNetMATHCrossRef

32.

Zhang, C., Wang, H., Huang, J.F., Wang, C., Yue, X.Y.: A second order operator splitting numerical scheme for the “good” Boussinesq equation. Appl. Numer. Math. 119, 179–193 (2017)MathSciNetMATHCrossRef

33.

Lubich, C.: On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77, 2141–2153 (2008)MathSciNetMATHCrossRef

34.

Thalhammer, M.: Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50, 3231–3258 (2012)MathSciNetMATHCrossRef

35.

Zhai, S.Y., Wang, D.L., Weng, Z.F., Zhao, X.: Error analysis and numerical simulations of Strang splitting method for space Fractional nonlinear Schrödinger equation. J. Sci. Comput. 81, 965–989 (2019)MathSciNetMATHCrossRef

36.

Lee, H.G., Shin, J., Lee, J.Y.: First and second order operator splitting methods for the phase field crystal equation. J. Comput. Phys. 299, 82–91 (2015)MathSciNetMATHCrossRef

37.

Shen, J., Tang, T., Wang, L.L.: Spectral Methods Algorithms: Analyses and Applications, 1st edn. Springer, Berlin (2010)

38.

Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67, 73–85 (1998)MathSciNetMATHCrossRef

39.

Mishra, S., Svärd, M.: On stability of numerical schemes via frozen coefficients and the magnetic induction equations. BIT Numer. Math. 50, 85–108 (2010)MathSciNetMATHCrossRef

40.

Tadmor, E.: The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23, 1–10 (1986)MathSciNetMATHCrossRef

41.

Canuto, C., Quarteroni, A., Hussaini, M.Y., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)MATHCrossRef

42.

Gottlieb, S., Wang, C.: Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers’ Equation. J. Sci. Comput. 53, 102–128 (2012)MathSciNetMATHCrossRef

43.

Gottlieb, S., Tone, F., Wang, C., Wang, X., Wirosoetisno, D.: Long time stability of a classical efficient scheme for two-dimensional Navier–Stokes equations. SIAM J. Numer. Anal. 50, 126–150 (2012)MathSciNetMATHCrossRef

Titel: Stability and Error Estimate of the Operator Splitting Method for the Phase Field Crystal Equation
verfasst von: Shuying Zhai
Zhifeng Weng
Xinlong Feng
Yinnian He
Publikationsdatum: 01.01.2021
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2021
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01386-8

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