Kinetic equations model the position-velocity distribution of particles subject to transport and collision effects. Under a diffusive scaling, these combined effects converge to a diffusion equation for the position density in the limit of an infinite collision rate. Despite this well-defined limit, numerical simulation is expensive when the collision rate is high but finite, as small time steps are then required. In this work, we present an asymptotic-preserving multilevel Monte Carlo particle scheme that makes use of this diffusive limit to accelerate computations. In this scheme, we first sample the diffusive limiting model to compute a biased initial estimate of a Quantity of Interest, using large time steps. We then perform a limited number of finer simulations with transport and collision dynamics to correct the bias. The efficiency of the multilevel method depends on being able to perform correlated simulations of particles on a hierarchy of discretization levels. We present a method for correlating particle trajectories and present both an analysis and numerical experiments. We demonstrate that our approach significantly reduces the cost of particle simulations in high-collisional regimes, compared with prior work, indicating significant potential for adopting these schemes in various areas of active research.
翻译:动因方程式模拟受迁移和碰撞影响的颗粒的定位速度分布。 在分流缩放下, 这些合并效应会聚集到无限碰撞率极限中位置密度的折射方程中。 尽管有这一明确界定的极限, 当碰撞率高时, 数字模拟费用很高, 但因需要时序小而有限。 在这项工作中, 我们提出了一个无症状- 保存多级蒙特卡洛粒子方案, 利用这种分流限制来加速计算。 在这个办法中, 我们首先抽样采用 diffive 限制模型, 用大时间步骤计算偏差的初步利益量估计值。 然后我们用运输和碰撞动力进行数量有限的微小模拟, 以纠正偏差。 多层次方法的效率取决于能否在离散程度的层次上对颗粒进行相关模拟。 我们提出了一种将粒子轨迹联系起来的方法, 并同时提出分析和数字实验。 我们证明, 我们的方法大大降低了高分流系统粒子模拟的成本, 与先前的工作相比, 表明在各种积极研究领域采用这些计划的巨大潜力。