论文标题
与渐近保护多层次蒙特卡洛的扩散极限附近的玻尔兹曼-BGK方程的加速模拟
Accelerated simulation of Boltzmann-BGK equations near the diffusive limit with asymptotic-preserving multilevel Monte Carlo
论文作者
论文摘要
动力学方程模拟受运输和碰撞效应的颗粒的位置 - 速度分布。在扩散缩放的情况下，这些组合效应会收敛到无限碰撞速率极限的位置密度的扩散方程。尽管限制了这个定义明确的限制，但当碰撞速率高但有限时，数值模拟还是昂贵的，因为需要小时步。在这项工作中，我们提出了一种渐近保护多层次蒙特卡洛粒子方案，该方案利用这种扩散极限来加速计算。在此方案中，我们首先采样了扩散限制模型，以使用大的时间步骤计算一定数量的兴趣的初始估计。然后，我们对使用传输和碰撞动力学进行有限数量的较大模拟来纠正偏差。多级方法的效率取决于能够在离散水平的层次结构上执行粒子的相关模拟。我们提出了一种将粒子轨迹相关的方法，并提出了分析和数值实验。我们证明，与先前的工作相比，我们的方法大大降低了粒子模拟的成本，这表明在各个主动研究领域采用这些方案的显着潜力。
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.
