Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System

Chen, Wenbin1; Wang, Cheng2; Wang, Shufen3; Wang, Xiaoming4,5; Wise, Steven M.6

2020-07-20

10.1007/s10915-020-01276-z

JOURNAL OF SCIENTIFIC COMPUTING   影响因子和分区

0885-7474

1573-7691

We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. To overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, and this, in turn, yields a convex-concave decomposition of the physical energy in the ternary system. Consequently, both the unique solvability and the unconditional energy stability of the proposed numerical scheme are established at a theoretical level. In addition, an optimal rate convergence analysis in the l(infinity)8 (0, TN-1) boolean AND l(2)(0, T; H-N(1)) norm is provided, with Fourier pseudo-spectral discretization in space, which is the first such result in this field. To deal with the nonlinear implicit equations at each time step, we apply an efficient preconditioned steepest descent (PSD) algorithm. A second order accurate, modified BDF scheme is also discussed. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme.

Ternary Cahn-Hilliard system Convexity analysis Energy stability Optimal rate convergence analysis Fourier pseudo-spectral approximation Partial and total spreading

SCI ; EI

英语

其他

NSFC[11871159][11671098][91630309] ; 111 Project[B08018] ; NSF[DMS-1719854][DMS-1418689][DMS-1715504] ; Guangdong Key Laboratory[2019B030301001]

Mathematics

Mathematics, Applied

WOS:000553484700002

SPRINGER/PLENUM PUBLISHERS

20203008969141

Optimization ; Phase space methods ; System stability ; Nonlinear equations ; Nonlinear analysis

Thermodynamics:641.1 ; Mathematics:921 ; Optimization Techniques:921.5 ; Systems Science:961

MATHEMATICS

Web of Science

1.Fudan Univ, Sch Math Sci, Shanghai Key Lab Contemporary Appl Math, Shanghai 200433, Peoples R China
2.Univ Massachusetts, Math Dept, N Dartmouth, MA 02747 USA
3.Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China
4.Southern Univ Sci & Technol, Dept Math, Shenzhen 518055, Peoples R China
5.Southern Univ Sci & Technol, SUSTech Int Ctr Math, Shenzhen 518055, Peoples R China
6.Univ Tennessee, Math Dept, Knoxville, TN 37996 USA

Chen, Wenbin,Wang, Cheng,Wang, Shufen,et al. Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System[J]. JOURNAL OF SCIENTIFIC COMPUTING,2020,84(2).

Chen, Wenbin,Wang, Cheng,Wang, Shufen,Wang, Xiaoming,&Wise, Steven M..(2020).Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System.JOURNAL OF SCIENTIFIC COMPUTING,84(2).

Chen, Wenbin,et al."Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System".JOURNAL OF SCIENTIFIC COMPUTING 84.2(2020).