两类相场方程的紧致指数时间差分法设计

COMPACT EXPONENTIAL TIME DIFFERENCE METHODS FOR TWO PHASE FIELD MODELS

本文针对相场方程提出稳定的高阶紧致指数时间差分算法.该算法具有完全显式的特性,从而避免了求解线性或非线性方程组.算法使用精确指数时间差分和多步法近似以保证精确性;通过线性算子分裂控制刚性非线性项以增强稳定性;同时引入有限差分格式的紧致表示大大降低了指数时间差分法的存储需求和计算量.算法的精确性和高效性通过CahnHilliard方程和Willmore问题相场模型的大规模三维模拟进行了验证.

In this work, we develop stable high order compact exponential time differencing method for solving phase field equations. The proposed method is completely explicit in nature, thus free from the need to solve linear and nonlinear systems. It utilizes accurate exponential time differences with multistep approximations to maintain accuracy. Novel linear opera- tor splitting techniques are incorporated to control the stiff nonlinear terms and improve the discrete energy stability. Furthermore, compact representation of central difference is used for discretizations of spatial operators so that the storage and computation costs are reduced significantly. Large scale three-dimensional simulations for Cahn-Hilliard equation and phase field model of the Willmore problem are presented to demonstrate the accuracy and effectiveness of the proposed method.