Heriot-Watt Mathematics Report Series
HWM99-6, 10 Mar 1999

Approximating the Becker-Döring cluster equations

D B Duncan and A R Soheili


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 metatstable 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 behaviour of the problem with a much reduced system of equations. Our schemes are based on a piecewise constant flux approximation, a Galerkin method using a discrete inner product, and a discretisation of a PDE that in turn approximates the Becker-Döring equations. We establish a posteriori error estimates for two of the schemes and report on the results of a representative set of numerical experiments. All three schemes give good results, and the PDE scheme appears to be more efficient than the others.

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