A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation
A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation
复制标题
DOI：
10.1007/bf01055714
复制
发表时间：
1992-02
影响因子：
1.6
通讯作者：
W. Wagner
中科院分区：
文献类型：
--
作者：
W. Wagner
考虑波尔兹曼方程的伯德直接模拟蒙特卡罗方法。与Bird算法相关的随机经验测量的极限(当粒子数趋于无穷大时)被证明是一个确定性测值函数，它满足一个方程，该方程在某种意义上接近于Boltzmann方程。介绍了一个马尔可夫跳跃过程，该过程通过一个随机时间变换与Bird的碰撞模拟过程相联系。建立了马尔可夫过程和随机时间变换的收敛性。这些结果，再加上一些关于随机测度收敛的一般性质，使得刻画Bird算法的极限行为成为可能。
Bird's direct simulation Monte Carlo method for the Boltzmann equation is considered. The limit (as the number of particles tends to infinity) of the random empirical measures associated with the Bird algorithm is shown to be a deterministic measure-valued function satisfying an equation close (in a certain sense) to the Boltzmann equation. A Markov jump process is introduced, which is related to Bird's collision simulation procedure via a random time transformation. Convergence is established for the Markov process and the random time transformation. These results, together with some general properties concerning the convergence of random measures, make it possible to characterize the limiting behavior of the Bird algorithm.