空间分数阶质量守恒型Allen-Cahn方程的高效数值算法研究

项目名称： 空间分数阶质量守恒型Allen-Cahn方程的高效数值算法研究

项目编号： No.11526094

项目类型： 专项基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 翟术英

作者单位： 华侨大学

项目金额： 3万元

中文摘要： 空间分数阶守恒型Allen-Cahn方程是材料学中研究相变理论和界面动力学的基本模型。但由于分数阶导算子及非局部守恒项的影响，使得许多求解经典整数阶Allen-Cahn方程行之有效的数值方法在解决此类问题时遇到了严重困难。本项目针对空间分数阶守恒型Allen-Cahn方程，研究其数值逼近中的高性能数值算法。首先利用算子分裂将原问题分裂为三个简单子问题，然后分别采用谱方法、解析法、高阶紧致差分方法建立每个子问题的离散格式。并运用能量方法严格分析算法的能量不增和质量守恒性。然后构造合理有效的自适应估计子，对时间步长进行自适应处理。该项目的研究成果将加深对反常扩散过程本质的理解，并为非线性科学的研究和发展及复杂动力学行为的研究提供新途径。

中文关键词： 空间分数阶Allen-Cahn方程；空间分数阶 Cahn-Hilliard 方程；算子分裂方法；紧致差分方法；谱方法

英文摘要： Mass conserving Allen-Cahn equation with space fractional derivatives is a basic model for phase transitions and interfacial dynamics in materials science. However, because of the influence of the fractional operator and the nonlocal conserving term, the effective methods for solving classical integer order problems are not valid any more. In this project, we focus on devising the efficient algorithm for the mass conserving Allen-Cahn equation with space fractional derivatives. Firstly, based on operator splitting method, the original problem is divided into three simple sub-equations, then spectral method and analytical method and high order compact difference scheme are used to solve them, respectively. Meanwhile, the energy degradation and the mass conservation of the proposed method will be proved rigorously by the energy method. The research results in this project are essential to expanding understanding the essence of anomalous diffusion process, and will provide the new methods and procedures to develop the nonlinear scientific research and complex dynamic behaviors research.

英文关键词： Space fractional Allen-Cahn equation；Space fractional Cahn-Hilliard equation；Operator-splitting method；Compact difference scheme；Spectral method