相场方程的高效数值算法

Abstract

本文回顾求解相场方程数值方法的一些最新进展. 数值求解相场方程的主要难点在于非线性项 和高阶微分项对时间步长有严格限制, 而相场方程的数值模拟通常需要很长的计算时间才能达到稳定 状态. 众所周知, 相场模型满足一种称为能量稳定的非线性稳定关系, 通常表示为自由能泛函随时间 递减. 如何设计满足离散能量稳定的数值格式, 使得可以进行大时间步长同时又准确地模拟, 近来越 来越受到重视. 本文将针对一些常见的相场方程阐述几类广泛使用的高效数值格式, 以及基于能量随 时间的变化率而设计的一种时间自适应算法, 使得数值解的准确性和算法稳定性得到保证的前提下, 计算效率大大提高. In this article, we overview recent developments of numerical methods for phase-field equations. The main difficulty for numerically solving phase-field equations is about a severe restriction on the time step due to nonlinearity and high order differential terms, while it usually requires a very long time simulation to reach the steady state. It is known that phase-field models satisfy a nonlinear stability relationship, called energy stability, which means that the free energy functional decays in time. It has attracted more and more attention to design numerical schemes inheriting the energy stability so that the numerical simulation may use large time steps and keep the accuracy. For some popularly studied phase-field equations, this article will present several widely used highly efficient numerical schemes and show an adaptive time-stepping strategy based on the changing rate in time of the energy functional, which could guarantee the accuracy and stability of the numerical solution and improves the computational efficiency significantly.