Controlling the accuracy of unconditionally stable algorithms in Cahn-Hilliard Equation

TL;DR

This paper develops a method to control and improve the accuracy of unconditionally stable algorithms for the Cahn-Hilliard equation by analytically relating algorithmic and physical time steps, leading to an optimized driving scheme.

Contribution

It introduces a general calculation for the analytic time step in terms of the algorithmic time step, enhancing accuracy control in unconditionally stable algorithms for the Cahn-Hilliard equation.

Findings

Derived an improved driving scheme $ ext{d} au=At^{2/3}$

Validated the scheme through numerical experiments

Confirmed previous numerical observations

Abstract

Given an unconditionally stable algorithm for solving the Cahn-Hilliard equation, we present a general calculation for an analytic time step $\d \tau$ in terms of an algorithmic time step $\dt$. By studying the accumulative multi-step error in Fourier space and controlling the error with arbitrary accuracy, we determine an improved driving scheme $\dt=At^{2/3}$ and confirm the numerical results observed in a previous study \cite{Cheng1}.