基于核的方法的非完全混合化学反应的拉格朗日方法
A Kernel-based Lagrangian Method for Imperfectly-mixed Chemical Reactions
AI总结：
本文提出基于核的拉格朗日方法用于模拟非完全混合化学反应，通过核函数减少粒子数量，提高计算效率。
摘要：
当前用于模拟扩散-反应方程的拉格朗日（粒子追踪）算法必须使用一定数量的粒子以准确模拟系统动态，特别是对于非完全混合系统。粒子数量与系统初始浓度场的统计特性相关。具有短程相关性和/或小浓度方差的系统需要更多粒子，这可能限制方法的计算可行性。在著名的双分子反应问题中，我们证明使用基于核的粒子而非狄拉克delta粒子可以显著减少所需粒子数。我们推导出给定减少粒子数的高斯核固定宽度，从而在任何指定时间消除核与狄拉克解之间的误差。我们还展示了如何通过最小化解的平方差异来求解固定核尺寸。数值结果表明，核的宽度应保持在域大小的约12%以下，并且用于推导核宽度的解析方程因忽略了高阶矩而显著受损。通过最小二乘法确定的核宽度的模拟表现优于那些在特定时间匹配的模拟。基于先前结果的启发式时间可变核尺寸与最小二乘固定核尺寸的性能相当。
英文摘要：
Current Lagrangian (particle-tracking) algorithms used to simulate diffusion-reaction equations must employ a certain number of particles to properly emulate the system dynamics---particularly for imperfectly-mixed systems. The number of particles is tied to the statistics of the initial concentration fields of the system at hand. Systems with shorter-range correlation and/or smaller concentration variance require more particles, potentially limiting the computational feasibility of the method. For the well-known problem of bimolecular reaction, we show that using kernel-based, rather than Dirac-delta, particles can significantly reduce the required number of particles. We derive the fixed width of a Gaussian kernel for a given reduced number of particles that analytically eliminates the error between kernel and Dirac solutions at any specified time. We also show how to solve for the fixed kernel size by minimizing the squared differences between solutions over any given time interval. Numerical results show that the width of the kernel should be kept below about 12% of the domain size, and that the analytic equations used to derive kernel width suffer significantly from the neglect of higher-order moments. The simulations with a kernel width given by least squares minimization perform better than those made to match at one specific time. A heuristic time-variable kernel size, based on the previous results, performs on a par with the least squares fixed kernel size.
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