Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation
Tóm tắt
The Swift–Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift–Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in Liu and Yin (J Sci Comput 77:467–501, 2018) for the spatial discretization, and the “Invariant Energy Quadratization” method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.
Tài liệu tham khảo
Badia, S., Guillen-Gonzalez, F., Gutierrez-Santacreu, J.V.: Finite element approximation of nematic liquid crystal flows using a saddle-point structure. J. Comput. Phys. 230, 1686–1706 (2011)
Bangerth, W., Hartmann, R., Kanschat, G.: deal.II–A general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007)
Braaksma, B., Iooss, G., Stolovitch, L.: Proof of quasipatterns for the Swift–Hohenberg equation. Commun. Math. Phys. 353, 37–67 (2017)
Christov, C.I., Pontes, J.: Numerical scheme for Swift–Hohenberg equation with strict implementation of Lyapunov functional. Math. Comput. Model. 35, 87–99 (2002)
Christov, C.I., Pontes, J., Walgraef, D., Velarde, M.G.: Implicit time splitting for fourth-order parabolic equations. Comput. Methods Appl. Mech. Eng. 148, 209–224 (1997)
Cross, M., Greenside, H.: Pattern Formation and Dynamics in Nonequilibrium Systems. Cambridge University Press, New York (2009)
Dee, G., Saarloos, W.: Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett. 60, 2641–2644 (1988)
Dehghan, M., Abbaszadeh, M.: The meshless local collocation method for solving multi-dimensional Cahn–Hilliard, Swift–Hohenberg and phase field crystal equations. Eng. Anal. Bound. Elem. 78, 49–64 (2017)
Evstigneev, N.M., Magnitskii, N.A., Sidorov, S.V.: Nonlinear dynamics of laminar-turbulent transition in three dimensional Rayleigh–Bénard convection. Commun. Nonlinear Sci. Numer. Simul. 15, 2851–2859 (2010)
Fife, P.C., Kowalczyk, M.: A class of pattern-forming models. J. Nonlinear Sci. 9, 641–669 (1999)
Gomez, H., Nogueira, X.: A new space-time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional. Commun. Nonlinear Sci. Numer. Simul. 17(12), 4930–4946 (2012)
Guillen-Gonzalez, F., Tierra, G.: On linear schemes for a Cahn–Hilliard diffuse interface model. J. Comput. Phys. 234, 140–171 (2013)
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, 1st edn. Springer, Berlin (2007)
Hoyle, R.B.: Pattern Formation: An Introduction to Methods. Cambridge University Press, Cambridge (2006)
Hutt, A., Atay, F.M.: Analysis of nonlocal neural fields for both general and gamma-distributed connectivities. Physica D 203, 30–54 (2005)
Kudryashov, N.A., Sinelshchikov, D.I.: Exact solutions of the Swift–Hohenberg equation with dispersion. Commun. Nonlinear Sci. Numer. Simul. 17, 26–34 (2012)
Lee, H.G.: A semi-analytical Fourier spectral method for the Swift–Hohenberg equation. Comput. Math. Appl. (CMA) 74(8), 1885–1896 (2017)
Liu, H., Yin, P.: A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems. J. Sci. Comput. 77, 467–501 (2018)
Peletier, L.A., Rottschäfer, V.: Pattern selection of solutions of the Swift–Hohenberg equation. Physica D Nonlinear Phenom. 194(1), 95–126 (2004)
Peletier, L.A., Troy, W.C.: Spatial patterns described by the extended Fisher–Kolmogorov (EFK) equation: kinks. Differ. Integral Equ. 8, 1279–1304 (1995)
Pérez-Moreno, S.S., Chavarría, S.R., Chavarría, G.R.: Numerical solution of the Swift–Hohenberg equation. In: Klapp, J., Medina, A. (eds.) Experimental and computational fluid mechanics. Environmental science and engineering, pp. 409–416. Springer, Cham (2014)
Riviére, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Society for Industrial and Applied Mathematics, Philadelphia (2008)
Sarmiento, A.F., Espath, L.F.R., Vignal, P., Dalcin, L., Parsani, M., Calo, V.M.: An energy-stable generalized-\(\alpha \) method for the Swift–Hohenberg equation. J. Comput. Appl. Math. 344, 836–851 (2018)
Shen, J., Xu, J., Yang, X.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)
Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability. Numerical solutions of partial differential equations. In: Bertoluzza, S., Falletta, S., Russo, G., Shu, C.-W. (eds.) Advanced Courses in Mathematics CRM Barcelona, pp. 149–201. Birkhauser, Basel (2009)
Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977)
Thiele, U., Archer, A.J., Robbins, M.J., Gomez, H., Knobloch, E.: Localized states in the conserved Swift–Hohenberg equation with cubic nonlinearity. Phys. Rev. E 87, 042915 (2013)
van den Berg, G.J.B., Peletier, L.A., Troy, W.C.: Global branches of multi-bump periodic solutions of the Swift–Hohenberg equation. Arch. Ration. Mech. Anal. 158, 91–153 (2001)
Wen, B., Dianati, N., Lunasin, E., Chini, G.P., Doering, C.R.: New upper bounds and reduced dynamical modeling for Rayleigh–Bénard convection in a fluid saturated porous layer. Commun. Nonlinear Sci. Numer. Simul. 17(5), 2191–2199 (2012)
Xi, H., Viñals, J., Gunton, J.D.: Numerical solution of the Swift–Hohenberg equation in two dimensions. Physica A 177, 356–365 (1991)
Yang, X.: Linear, first and second order and unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 302, 509–523 (2016)
Zhang, Z., Ma, Y.: On a large time-stepping method for the Swift–Hohenberg equation. Adv. Appl. Math. Mech. 8, 992–1003 (2016)
Zhao, J., Wang, Q., Yang, X.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int. J. Numer. Methods Eng. 110(3), 279–300 (2017)