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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
29/2006

On the Efficient Evaluation of Coalescence Integrals in Population Balance Models

Wolfgang Hackbusch

Abstract

The solution of population balance equations is a function $f(t,r,x)$ describing a population density of particles of the property $x$ at time $t$ and space $r.$ For instance, the additional independent variable $x$ may denote the particle size. The describing partial differential equation contains additional sink and source terms involving integral operators. Since the coordinate $x$ adds at least one further dimension to the spatial directions and time coordinate, an efficient numerical treatment of the integral terms is crucial. One of the more involved integral terms appearing in population balance models is the coalescence integral, which is of the form $\int_{0}^{x}\kappa(x-y,y)f(y)f(x-y)\mathrm{d}y.$ In this paper we describe an evaluation method of this integral which needs only $\mathcal{O} (n\log n)$ operations, where $n$ is the number of degrees of freedom with respect to the variable $x.$ This cost can also be obtained in the case of a grid geometrically refined towards $x=0$.

Received:
Mar 16, 2006
Published:
Mar 16, 2006
MSC Codes:
45E99, 45K05, 92D25
Keywords:
population balance model, aggregation, agglomeration, coalescence, convolution integral, integro-partial differential equation

Related publications

inJournal
2006 Repository Open Access
Wolfgang Hackbusch

On the efficient evaluation of coalescence integrals in population balance models

In: Computing, 78 (2006) 2, pp. 145-159