A Kernel-based Lagrangian Method for Imperfectly-mixed Chemical Reactions
For researchers simulating diffusion-reaction systems, this offers a way to reduce computational cost while maintaining accuracy, though the improvement is incremental.
The paper tackles the computational bottleneck of Lagrangian particle-tracking methods for imperfectly-mixed chemical reactions, where many particles are needed. Using kernel-based particles (Gaussian) instead of Dirac-delta reduces particle count; a least-squares optimized kernel width performs best, with a heuristic time-variable kernel matching its performance.
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.