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Analysis of the operator splitting scheme for the Cahn‐Hilliard equation with a viscosity term
Author(s) -
Weng Zhifeng,
Zhai Shuying,
Feng Xinlong
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22378
Subject(s) - cahn–hilliard equation , mathematics , operator (biology) , mathematical analysis , nonlinear system , operator splitting , term (time) , norm (philosophy) , convergence (economics) , conservation of mass , viscosity solution , partial differential equation , physics , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene , political science , law , economics , economic growth
In this paper, we consider a second‐order fast explicit operator splitting method for the viscous Cahn‐Hilliard equation, which includes a viscosity term α Δ u t ( α ∈ (0, 1)) described the influences of internal micro‐forces. The choice α = 0 corresponds to the classical Cahn‐Hilliard equation whilst the choice α = 1 recovers the nonlocal Allen‐Cahn equation. The fundamental idea of our method is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using a pseudo‐spectral method, and thus an ordinary differential equation is obtained. The nonlinear one is solved via TVD‐RK 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. Besides, a detailed comparison is made for the dynamics and the coarsening process of the metastable pattern for various values of α . Moreover, energy degradation and mass conservation are also verified.