Why large time-stepping methods for the Cahn-Hilliard equation is stable

Dong Li
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引用次数: 3

Abstract

We consider the Cahn-Hilliard equation with standard double-well potential. We employ a prototypical class of first order in time semi-implicit methods with implicit treatment of the linear dissipation term and explicit extrapolation of the nonlinear term. When the dissipation coefficient is held small, a conventional wisdom is to add a judiciously chosen stabilization term in order to afford relatively large time stepping and speed up the simulation. In practical numerical implementations it has been long observed that the resulting system exhibits remarkable stability properties in the regime where the stabilization parameter is O(1), the dissipation coefficient is vanishingly small and the size of the time step is moderately large. In this work we develop a new stability theory to address this perplexing phenomenon.
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为什么大时间步进法求解Cahn-Hilliard方程是稳定的
我们考虑具有标准双阱势的Cahn-Hilliard方程。采用一类典型的一阶时间半隐式方法,对线性耗散项进行隐式处理,对非线性项进行显式外推。当耗散系数保持较小时,传统的做法是添加一个明智选择的稳定项,以提供相对较大的时间步进并加快模拟速度。在实际的数值实现中,人们长期观察到,在稳定参数为0(1)、耗散系数很小、时间步长适中的情况下,所得到的系统表现出显著的稳定性。在这项工作中,我们发展了一个新的稳定性理论来解决这个令人困惑的现象。
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