| 付航宇,韩涛,D. M. McFarland,程相乐,卢奂采.Fokker-Planck 方程的算子分裂算法[J].高技术通讯(中文),2024,34(10):1070~1080 |
| Fokker-Planck 方程的算子分裂算法 |
| Operator splitting method for Fokker-Planck equation |
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| DOI：10. 3772 / j. issn. 1002-0470. 2024. 10. 006 |
| 中文关键词: 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 |
| 英文关键词: finite element method, Fokker-Planck (FP) equation, operator splitting, nonlinear Duffing system, random vibration |
| 基金项目: |
| 作者 | 单位 | | 付航宇 | (浙江工业大学机械工程学院声学与振动实验室 杭州 310014) | | 韩涛 | | | D. M. McFarland | | | 程相乐 | | | 卢奂采 | |
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| 中文摘要: |
| 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。 |
| 英文摘要: |
| 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 oper-
ator splitting and finite element method, not only high numerical accuracy, but also fast solution speed. The opera-
tor splitting method which is divided into convection term and diffusion term is the best way. The experimental re-
sults 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 nu-
merical accuracy and computational efficiency, and it is a potential numerical solution of FP equation. |
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