2016/10/08 by Wenqiang Feng, Zhen Guan, Feng, Wenqiang +7
Earth and Planetary Sciences · Engineering · Materials Science · #35K35 #35K55 #65M06 #65M12 #FOS: Mathematics #Fluid Dynamics and Thin Films #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena #nanoparticles nucleation surface interactions
paper · pdf · doi:10.48550/arxiv.1610.02473
openalex publication_date 2016/10/08 · openalex created_date 2016/10/21 · openalex updated_date 2026/07/28
We present and analyze an unconditionally energy stable and convergent finite difference scheme for the Functionalized Cahn-Hilliard equation. One key difficulty associated with the energy stability is based on the fact that one nonlinear energy functional term in the expansion appears as non-convex, non-concave. To overcome this subtle difficulty, we add two auxiliary terms to make the combined term convex, which in turns yields a convex-concave decomposition of the physical energy. As a result, an application of the convex splitting methodology assures both the unique solvability and the unconditional energy stability of the proposed numerical scheme. To deal with a 4-Laplacian solver in an H-1 gradient flow at each time step, we apply an efficient preconditioned steepest descent algorithm to solve the corresponding nonlinear systems. In addition, a global in time H\rm per2 stability of the numerical scheme is established at a theoretical level, which in turn ensures the full order convergence analysis of the scheme. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme.