An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

TL;DR

This paper develops an unconditionally energy stable finite difference scheme for the stochastic Cahn-Hilliard equation, incorporating convex splitting, Newton iteration, and adaptive time stepping for efficient long-term simulations.

Contribution

It introduces a novel unconditionally energy stable scheme for the stochastic Cahn-Hilliard equation using convex splitting and adaptive time stepping.

Findings

The scheme is unconditionally energy stable and uniquely solvable.

Adaptive time stepping improves efficiency in long-term simulations.

Numerical experiments confirm stability, efficiency, and stochastic effects.

Abstract

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.