We develop a novel Monte Carlo strategy for the simulation of the Boltzmann-BGK model with both low-collisional and high-collisional regimes present. The presented solution to maintain accuracy in low-collisional regimes and remove exploding simulation costs in high-collisional regimes uses hybridized particles that exhibit both kinetic behaviour and diffusive behaviour depending on the local collisionality. In this work, we develop such a method that maintains the correct mean, variance, and correlation of the positional increments over multiple time steps of fixed step size for all values of the collisionality, under the condition of spatial homogeneity during the time step. In the low-collisional regime, the method reverts to the standard velocity-jump process. In the high-collisional regime, the method collapses to a standard random walk process. We analyze the error of the presented scheme in the low-collisional regime for which we obtain the order of convergence in the time step size. We furthermore provide an analysis in the high-collisional regime that demonstrates the asymptotic-preserving property.

翻译：我们为模拟Boltzmann-BGK模型开发了一个新型的蒙特卡洛战略,模拟Boltzmann-BGK模型,该模型同时具有低等级和高等级制度;为保持低等级制度的准确性和消除高等级制度中的爆炸性模拟成本而提出的解决方案,使用根据局部碰撞性而表现出动能行为和悬浮行为的混合颗粒;在这项工作中,我们开发了这样一种方法,在时间步骤空间同质性条件下,保持碰撞所有数值在固定步骤大小的多个时间步骤上的方位递增的正确平均值、差异和相关性;在低等级制度中,该方法恢复到标准速度跳跃过程;在高等级制度中,该方法崩溃到标准的随机行走过程;我们分析了低等级制度中所提出的办法的错误,我们为此在时间步骤大小上取得了趋同的顺序。我们还对高等级制度进行了分析,以显示保留财产的高度速度。