Approximation of coalescence integrals in population balance models with local mass conservation

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 mass of the particle. The describing 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 $$\int_{0}^{x} \kappa(x-y,y)f(y) f(x-y)\mathrm{d}y.$$ The discretisation may use a locally refined grid. In this paper we describe an algorithm which (i) is efficient (the cost is $$\mathcal{O}(N\log N)$$, N: data size) and (ii) ensures local mass conservation.