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MiS Preprint

On the Efficient Evaluation of Coalescence Integrals in Population Balance Models

Wolfgang Hackbusch


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 $x=0$.

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

Related publications

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