-matrix methods for linear and quasi-linear integral operators appearing in population balances
Introduction
The description of a population involves a density function . This function represents the mean number of individuals with a special property at a certain time. In this study, we consider the situation that each individual is fully identified by its volume . The function describes the dynamic of the population through a systemThe term “…” indicates the possibility to add several source and sink terms (cf. Ramkrishna, 2000). In this work, we consider only the operators and which model breakage and coagulation phenomena, respectively.
The operator is the sum of operators
models the source that appears if an individual breaks into two or more smaller individuals. Here, the average amount of newly build individuals is denoted by , the breakage rate is given by and the size distribution by . The breaking individual vanishes from the population which is modelled by the sink term , where is again the breakage rate.
In a similar way, the operator consists of the summandsThe operator models the event that an individual is newly formed by coagulation of two individuals with volume and volume , respectively, whereas the sink term is motivated by the fact that every coagulating individual is vanishing from the population. Function models the coagulation event. A more detailed description of the operators and can be found, e.g. in Coulaloglou and Tavlarides (1977) or Gerstlauer (1999).
In this article, we provide an efficient numerical approach for the matrices associated to the operators (2a) and (3b). We consider, in this work, the following kernel functions:
- (1)
Breakage kernels(for the linear integral operator)
- •
emulsion kernel (Coulaloglou & Tavlarides, 1977)where
- •
- (2)
Coagulation kernels (for the quasi-linear integral operator)
- •
Smoluchowski kernel (Smoluchowski (1917))
- •
modified Smoluchowski kernel
- •
emulsion kernel (Coulaloglou & Tavlarides, 1977)
- •
To solve the system (1), a discretisation of the operators and is necessary. The behaviour of these operators depends mainly on the kernel functions and . For solving the system (1), usually an explicit scheme is used due to the integral character of the operators and . After discretisation, these operators are represented by (usual full) matrices and for the explicit solving a matrix–vector multiplication has to be performed for each time step.
For the numerical treatment of the quadratic operator , a first approach can be found in Koch (2005) and Hackbusch (2006).
In this paper, the solution of (1) is not considered as well as the effect of the so-called finite domain error, see, e.g. Attarakih, Bart, & Faqir, 2004.
Section snippets
Discretisation
Remark 1 Although the integration domain in (3b) is infinite, for numerical reasons a volume can be used as an upper limit. In the worst case, could be, e.g. the overall volume of the individuals. Moreover, without loss of generality, we assume . Then the domain of definition for the kernel functions is .
We rename the variables and denoting the property of the individuals, by x and y. Furthermore, the operators (2) and (3) do not depend on t therefore in the following
Separable kernel functions
The Smoluchowski kernel (5) shows a special property: it can be written in the form
as a sum of terms, which are products of factors with separated x and y dependence. Such a kernel function is called separable and can generally be written in the way
where k is called the separation rank.
Approximation error
The interpolation error of follows the estimates of Lemma 17 and Corollary 18 for (6) as well as Lemma 19 and Corollary 20 for (7). For (4), the interpolation error is given by the estimate of Lemma 22. Therefore, the error between the full matrix and the corresponding -matrix of order k is expected to behave as (exponential decreasing). In Fig. 8, the relative error is given as a function of k in the spectral norm and shows the predicted behaviour. The special
Conclusions
The discretisation of the linear and quasi-linear integral operator, (2a) and (3b), demands usually a complexity of . This circumstance makes the solving of the integro-differential equation (1) very expensive.
We have shown that for separable kernel functions it is possible to reduce this quadratic complexity to a linear one, i.e. to , even if the matrices, associated with the integral operators, are triangular. Furthermore, we introduced a new admissibility condition for the local
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