Convex Splitting Schemes Interpreted as Fully Implicit Schemes in Disguise for Phase Field Modeling

Convex splitting schemes (CSS in short) are among the most popular numerical schemes used in phase-field modeling. In this paper, we report a simple observation that some well-known CSS can be interpreted as some fully implicit schemes (FIS in short) in disguise. For the Allen-Cahn model, we prove that the standard CSS is exactly the same as the standard FIS but with a (much) smaller time step size and as a result, it would provide an approximation to the original solution of the Allen-Cahn model at a delayed time (although the magnitude of such a delay is reduced when the time step size is reduced). For the Cahn-Hilliard model, we prove that the standard CSS is exactly the same as the standard FIS for a different model that is a (nontrivial) perturbation of the original Cahn-Hilliard model. Motivated by such an equivalence between CSS and FIS, we propose a modification of a typical FIS or CSS for the Allen-Cahn so that the maximum principle will be valid on the discrete level and we further rigorously prove that, the linearization of such a modified FIS or CSS can be uniformly preconditioned by a Poisson-like operator.

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