Approximating the Becker–Döring cluster equations

Abstract

The Becker–Döring equations model the dynamics of coagulation and fragmentation of clusters of identical particles. The model is an infinite system of ordinary differential equations (ODEs) which specify the rates of change of the concentrations of r-particle clusters. For numerical computation the system is truncated at clusters of a finite size, but this might have to be prohibitively large to capture the metastable behaviour of the system. In this work we derive and investigate the properties of approximations of the Becker–Döring equations which aim to capture the metastable behaviour of the problem with a much reduced system of equations. The schemes are based on a piecewise constant flux approximation, a Galerkin method using a discrete inner product, a discretization of a PDE that in turn approximates the Becker–Döring equations and a lumped coefficients model. We establish a posteriori error estimates for three of these schemes and report on the results of a set of numerical experiments involving metastable behaviour in the solution. Three of the schemes give accurate results on a reduced set of equations, and the PDE scheme appears to be more efficient than the others for this problem.