强非线性、小参数、高阶偏微分方程的高效数值方法是科学计算领域的一个重大研究问题，涉及面广，模型众多。本项目以经典的Cahn-Hilliard模型方程为切入点，发展一类高阶无条件稳定、大时间步长、自适应的新型耦合数值方法，从格式构建、理论分析、计算实现三方面出发，主要研究：（1）Cahn-Hilliard方程的有限体积及紧致差分的IMEX-Runge-Kutta\Strong stability preserving稳定高阶方法的构造及理论分析；（2）Cahn-Hilliard方程的时空自适应方法的构造和理论分析。项目所提出的新方法具有时间步长大、稳定性强、计算量低、存储量小等优点，可以有效处理强非线性及小参数，可以满足大规模计算实际问题时对模拟精度和置信度的要求，实现高效高精度算法。

Efficient numerical method for strong nonlinear，small parameter, high-order partial differential equation is an important research field of scientific computing.This project is based on the classical Cahn-Hilliard model equation. A class of adaptive high-order methods of unconditionally stability, lager time-step will be proposed. The contents are as follows:(1) for the Cahn-Hilliard equation, the high-order lager time-step methods of FV\compact difference coupled with IMEX-Runge-Kutta\Strong stability preserving will be analyzed. (2) the space-time adaptive method will be given, and the theoretical analysis will be deduced. The methods in this project have some advantages: lager time-step, strong stability, low computational complexity, small storage capacity etc., and could effectively solve the problems with strong nonlinear, small parameter, high order partial differential equations. It might meet the simulation accuracy and reliability requirements of large-scale computing problems, and achieve high efficiency and high precision algorithm.