Smoluchowski aggregation-fragmentation equations: Fast numerical algorithm for steady-state solution

TL;DR

This paper introduces a novel, efficient numerical algorithm for quickly computing steady-state solutions of large aggregation-fragmentation systems described by Smoluchowski equations, significantly outperforming existing methods.

Contribution

The paper presents a new fast iterative algorithm for steady-state solutions of large Smoluchowski aggregation-fragmentation systems, applicable under mild kernel restrictions.

Findings

Algorithm is significantly faster than standard methods.

Applicable to large systems with steeply decreasing concentrations.

Works under mild restrictions on the kernel.

Abstract

We propose an efficient and fast numerical algorithm of finding a \emph{stationary} solution of large systems of aggregation-fragmentation equations of Smoluchowski type for concentrations of reacting particles. This method is applicable when the stationary concentrations steeply decreases with increasing aggregate size, which is fulfilled for the most important cases. We show that under rather mild restrictions, imposed on the kernel of the Smoluchowski equation, the following numerical procedure may be used: First, a complete solution for a relatively small number of equations (a "seed system") is generated and then the result is exploited in a fast iterative scheme. In this way the new algorithm allows to obtain a steady-state solution for rather large systems of equations, by orders of magnitude faster than the standard schemes.