Energy stable and large time-stepping methods for the Cahn–Hilliard equation

References

• U.M. Ascher, J. Ruuth, and R.J. Spiteri, Implicit–explicit Runge–Kutta method for time dependent partial differential equations, Appl. Numer. Math. 25 (1997), pp. 151–167. doi: 10.1016/S0168-9274(97)00056-1
   Web of Science ®Google Scholar
• U.M. Ascher, J. Ruuth, and T.R. Wetton, Implicit–explicit method for time dependent partial differential equations, SIAM. J. Numer. Anal. 32 (1995), pp. 797–823. doi: 10.1137/0732037
   Web of Science ®Google Scholar
• S.M. Choo, S.K. Chung, and K.I. Kim, Conservative nonlinear difference scheme for the Cahn–Hilliard equation-II, Comput. Math. Appl. 39 (2000), pp. 229–243. doi: 10.1016/S0898-1221(99)00326-0
   Web of Science ®Google Scholar
• J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system I: interfacial free energy, J. Chem. Phys. 28 (1958), pp. 258–267. doi: 10.1063/1.1744102
   Web of Science ®Google Scholar
• A. Cardone, Z. Jackiewicz, A. Sandu, and H. Zhang, Extrapolated IMEX Runge–Kutta methods, Math. Model. Anal. 19 (2014), pp. 18–43. doi: 10.3846/13926292.2014.892903
   Web of Science ®Google Scholar
• A. Cardone, Z. Jackiewicz, A. Sandu, and H. Zhang, Extrapolation-based implicit–explicit general linear methods, Numer. Algorithms 65 (2014), pp. 377–399. doi: 10.1007/s11075-013-9759-y
   Web of Science ®Google Scholar
• L.Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Commun. 108 (1998), pp. 147–158. doi: 10.1016/S0010-4655(97)00115-X
   Web of Science ®Google Scholar
• E.M. Constantinescu and A. Sandu, Extrapolated implicit-eExplicit time stepping, SIAM J. Sci. Comput. 31 (2010), pp. 4452–4477. doi: 10.1137/080732833
   Web of Science ®Google Scholar
• Q. Du and R.A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM. J. Numer. Anal. 28 (1994), pp. 1310–1322. doi: 10.1137/0728069
   Web of Science ®Google Scholar
• C.M. Elliott and D.A. French, A nonconforming finite element method for the two-dimensinal Cahn–Hilliard equations, SIAM. J. Numer. Anal. 26 (1989), pp. 884–903. doi: 10.1137/0726049
   Web of Science ®Google Scholar
• C.M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn–Hilliard equation, Math. Comput. 58 (1992), pp. 603–630. doi: 10.1090/S0025-5718-1992-1122067-1
   Web of Science ®Google Scholar
• D.J. Eyre, An unconditionally stable one-step scheme for gradient systems, 1998. Available at http://www.math.utah.edu/eyre/research/methods/stable.ps
   Google Scholar
• D.J. Eyre, Unconditionally gradient stable time marching the Cahn–Hilliard equation, 1998. Available at http://www.math.utah.edu/eyre/research/methods/ch_numer.ps
   Google Scholar
• D. Furihata, A stable and conservative finite difference scheme for the Cahn–Hilliard equation, Numer. Math. 87 (2001), pp. 675–699. doi: 10.1007/PL00005429
   Web of Science ®Google Scholar
• D. Furihata, T. Onda, and M. Mori, A numerical analysis of the Cahn–Hilliard equation by the finite difference scheme (in Japanese), Trans. Jpn. Soc. Ind. Appl. Math. 3 (1993), pp. 217–228.
   Google Scholar
• Y.N. He, Y.X. Liu, and T. Tang, On large time-stepping methods for the Cahn–Hilliard equation, Appl. Numer. Math. 57 (2007), pp. 616–628. doi: 10.1016/j.apnum.2006.07.026
   Web of Science ®Google Scholar
• M. Li and T. Tang, A compact fourth-order finite difference scheme for unsteady viscous incompressible flows, J. Sci. Comput. 16 (2001), pp. 29–45. doi: 10.1023/A:1011146429794
   Google Scholar
• M. Li, T. Tang, and B. Fornberg, A compact fourth-order finite difference scheme for the steady incompressible Navier–Stokes equations, Int. J. Numer. Methods 20 (1995), pp. 1137–1151. doi: 10.1002/fld.1650201003
   Web of Science ®Google Scholar
• E.V.L. de Mello and O.T. da Silveira Filho, Numerical study of the Cahn–Hilliard equation in one, two and three dimensions, Phys. A 347 (2005), pp. 429–443. doi: 10.1016/j.physa.2004.08.076
   Web of Science ®Google Scholar
• L. Pareschi and G. Russo, Implicit–explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25 (2005), pp. 129–155.
   Web of Science ®Google Scholar
• Z.H. Qiao, Z. Zhang, and T. Tang, An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM. J. Sci. Comput. 33 (2011), pp. 1395–1414. doi: 10.1137/100812781
   Web of Science ®Google Scholar
• Z.Z. Sun, A second-order accurate linearized difference scheme for the two-dimensional Cahn–Hilliard equation, Math. Comput. 64 (1995), pp. 1463–1471.
   Web of Science ®Google Scholar
• C.J. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM. J. Numer. Anal. 44 (2006), pp. 1759–1779. doi: 10.1137/050628143
   Web of Science ®Google Scholar
• H. Zhang, A. Sandu, and S. Blaise, Partitioned and implicit–explicit general linear methods for ODEs, J. Sci. Comput. 61 (2014), pp. 119–144. doi: 10.1007/s10915-014-9819-z
   Web of Science ®Google Scholar