Application of Monte Carlo method to solving boundary value problem of differential equations
-
摘要: 介绍了蒙特卡罗方法的基本原理以及随机数的产生方法。基于蒙特卡罗方法的思想，结合有限差分方法，建立了求解微分方程边值问题的随机概率模型，并以第一类边界条件的拉普拉斯方程和一个给定初值及边界条件的非稳态热传导方程为数值算例，研究了蒙特卡罗方法在求解微分方程边值问题中的应用。结果表明：利用蒙特卡罗方法，不仅可以有效解决给定边界条件的微分方程，对于给定初值条件的微分方程，也可以从时域有限差分方程出发，采用蒙特卡罗方法进行求解。数值模拟和对误差的理论分析均表明，增加蒙特卡罗试验中的模拟粒子点数，可以提高计算结果的精度。Abstract: This paper introduces the foundation of the Monte Carlo method and the way how to generate the random numbers. Based on the basic thought of the Monte Carlo method and the finite differential method, the stochastic model for solving the boundary value problem of differential equations is built. To investigate the application of the Monte Carlo method to solving the boundary value problem of differential equations, the model is used to solve Laplaces equations with the first boundary condition and the unsteady heat transfer equation with initial values and boundary conditions. The results show that the boundary value problem of differential equations can be effectively solved with the Monte Carlo method, and the differential equations with initial condition can also be calculated by using a stochastic probability model which is based on the time-domain finite differential equations. Both the simulation results and theoretical analyses show that the errors of numerical results are lowered as the number of simulation particles is increased.
-
点击查看大图
计量
- 文章访问数: 1814
- HTML全文浏览量: 248
- PDF下载量: 424
- 被引次数: 0
下载:
