A Kernel-based Lagrangian Method for Imperfectly-mixed Chemical Reactions

TL;DR

This paper introduces a kernel-based particle method for simulating imperfectly-mixed chemical reactions, reducing the number of particles needed and improving computational efficiency by analytically and numerically optimizing kernel size.

Contribution

It develops an analytical framework for selecting fixed and time-variable Gaussian kernel widths to minimize errors in particle-based reaction simulations.

Findings

Kernel-based particles significantly reduce required particle count.

Optimal kernel width should be below about 12% of domain size.

Least squares minimization improves kernel size selection over single-time matching.

Abstract

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.