Method of lines transpose: energy gradient flows using direct operator inversion for phase-field models. (English) Zbl 1407.65155
The authors illustrate how the method of lines transpose (MOL\(^{T}\)) can be used to solve nonlinear phase-field models, e.g. Cahn-Hilliard, vector Cahn-Hilliard, and functionalized Cahn-Hilliard equations. A novel factorization of the semidiscrete equation to ensure gradient stability, permitting large time steps is used. When combined with time adaptivity, it is possible to resolve rapid events such as spinodal evolution, which occur after long metastable states. The spatial solver is matrix-free, \(\mathcal{O}(N)\), and logically Cartesian, and the splitting error is directly incorporated into the nonlinear fixed point iterations. Higher-order time stepping, such as backward difference formulas, implicit Runge-Kutta, and spectral deferred correction methods is considered. It is found that the time adaptive backward Euler-backward difference 2 (BE-BDF2) method is the most efficient with respect to the accuracy and time to solution.
Reviewer: Abdallah Bradji (Annaba)
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M80 | Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 65M38 | Boundary element methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
method of lines transpose; Rothe’s method; implicit methods; boundary integral methods; alternating direction implicit methods; ADI schemes; unconditionally gradient stable schemes; Cahn-Hilliard; functionalized Cahn-HilliardReferences:
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