等离子体中分数阶微分方程求解的有限元方法研究

项目名称： 等离子体中分数阶微分方程求解的有限元方法研究

项目编号： No.11501441

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 郭士民

作者单位： 西安交通大学

项目金额： 18万元

中文摘要： 分数阶微分方程在等离子体中占有十分重要的地位，这是因为当介质具有非局部性质时，等离子体中的物理过程需要通过分数阶微分方程进行准确的描述。本项目综合运用计算数学、等离子体物理学和流体力学的知识，研究等离子体中分数阶微分方程求解的有限元方法，具体内容包括：针对高频—低频波发生耦合作用且具有非局部性质的等离子体模型，本项目将建立分数阶耦合非线性Schrödinger方程的有限元方法，对矢量非线性波动现象进行数值模拟，揭示分数阶导数对高频—低频波耦合作用的物理机理的影响；针对受到磁场作用且具有非局部性质的等离子体模型，本项目将建立分数阶MHD方程的多尺度有限元方法，对磁流体力学问题进行数值模拟，探索分数阶导数对多物理场耦合的作用规律。本项目将推动计算数学、等离子体物理学与流体力学的交叉发展，完善分数阶微分方程的数值算法，扩大分数阶微分方程的应用范围，为研究具有非局部性质的等离子体提供新的数学理论。

中文关键词： 有限元方法；分数阶微分方程；多尺度计算方法；等离子体

英文摘要： Fractional differential equations are of great importance in plasma, because the physical processes in the plasma can be accurately described by the fractional differential equations when the media possess the nonlocal properties. By using the computational mathematics, plasma physics, and fluid mechanics, this project will investigate the finite element methods for fractional differential equations in plasma. The project is composed of the following parts: For the plasma models with the coupling of high frequency-low frequency waves and nonlocal properties, it will establish the finite element methods for fractional coupled nonlinear Schrödinger equations, make the numerical simulations for the vector nonlinear waves, and reveal the effect of fractional derivative on the coupling of high frequency—low frequency waves; For the plasma models with the influence of magnetic fields and nonlocal properties, it will establish multiscale finite element methods for fractional MHD equations, make the numerical simulations for the magnetohydrodynamics problems, and explore the interactions between the fractional derivative and multiphysics coupling. This project will promote the cross development of computational mathematics, plasma physics, and fluid mechanics, improve the numerical methods for fractional differential equations, extend the applications of fractional differential equations, and provide new mathematical theories for investigating the plasma with nonlocal properties.

英文关键词： Finite element method;Fractional differential equations;Multiscale computation method;Plasma