Fokker-Planck方程的算子分裂算法  

Operator splitting method for Fokker-Planck equation

作　　者：付航宇 韩涛 D.M.McFarland 程相乐 卢奂采[1] FU Hangyu;HAN Tao;D.M.MCFARLAND;CHENG Xiangle;LU Huancai(Sound and Vibration Laboratory,College of Mechanical Engineering,Zhejiang University of Technology,Hangzhou 310014)

机构地区：[1]浙江工业大学机械工程学院声学与振动实验室,杭州310014

出　　处：《高技术通讯》2024年第10期1070-1080,共11页Chinese High Technology Letters

基　　金：国家自然科学基金(51975525)资助项目。

摘　　要：为减少传统Runge-Kutta有限单元法求解福克-普朗克(FP)方程所需的冗长时间,提出一种有限单元法结合算子分裂法的FP方程数值求解新方法。该方法通过对有限元矩阵方程的拆分,得到算子分裂子矩阵组,进而使用具有一阶和二阶精度的算子分裂法对FP方程进行数值求解。针对线性系统和非线性Duffing系统进行了FP方程数值求解的验证,检验了拆分为对流项和扩散项算子的计算精度和计算时间。实验结果表明,相对于传统的Runge-Kutta求解方法,在相同的数值解精度下,结合算子分裂法的求解时间仅为纯有限单元法的1%~5%。有限单元法结合算子分裂是一种具有较快速计算潜力的FP方程数值求解方法。In order to obtain the exact numerical solution of the Fokker-Planck(FP)equation efficiently,a numerical method which combines the finite element method and the operator splitting method is proposed in this paper.In this method,the FP equation is solved numerically by splitting the element matrix equation in space and then using the first-order and second-order accuracy operator splitting method.Numerical verification is carried out by using a linear system and a nonlinear Duffing system,and the results show that the method has the advantages of both operator splitting and finite element method,not only high numerical accuracy,but also fast solution speed.The operator splitting method which is divided into convection term and diffusion term is the best way.The experimental results show that compared with the Runge-Kutta method,the running time of the proposed method is only 1%-5%of the Runge-Kutta method.The finite element method combined with operator splitting has great advantages in numerical accuracy and computational efficiency,and it is a potential numerical solution of FP equation.

关 键 词：有限单元法 福克-普朗克(FP)方程 算子分裂 非线性Duffing系统 随机振动 

分 类 号：O241.6[理学—计算数学]