An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation
- 181 Downloads
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation, is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.
KeywordsCahn-Hilliard equation stochastic term energy stability convex splitting adaptive time stepping
MSC(2010)65M06 65M12 65Z05
Unable to display preview. Download preview PDF.
- 6.Chin J, Coveney P V. Lattice Boltzmann study of spinodal decomposition in two dimensions. Phys Rev E, 2002, 66: 016303-1–016303-8Google Scholar
- 9.Eyre D J. An unconditionally stable one-step scheme for gradient systems. Http://www.math.utah.edu/~eyre/ research/methods/stable.psGoogle Scholar
- 10.Eyre D J. Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Bullard J W, Kalia R, Stoneham M, et al., eds. Computational and Mathematical Models of Microstructural Evolution. Mater Res Soc Symp Proc, vol. 529. Warrendale: Materials Research Society, 1998, 39–46Google Scholar
- 11.Flory P J. Principles of Polymer Chemistry. New York: Cornell University Press, 1953Google Scholar
- 16.Huang J Y. Numerical study of the growth kinetics for TDGL equations. Master’s Degree Thesis. Ontario: The University of Western Ontario, 1998Google Scholar
- 23.Liu C H. Stochastic Process, 4th ed. Wuhan: Huazhong University of Science and Technology Press, 2008Google Scholar