H-matrix methods for quadratic integral operators appearing in population balances

Abstract In population dynamics the source term of breakage as well as the source and the sink term of coagulation are often described by an integral operator. All these operators are characterised by their kernel functions. Naive discretisation leads to full matrices and therefore to quadratic complexity O ( n 2 ) , if n denotes the number of degrees of freedom. In a recent paper [Koch, J., Hackbusch, W., & Sundmacher, K. (2007). H -matrix methods for linear and quasi-linear integral operators appearing in population balances. Computers & Chemical Engineering, 31 (7), 745–759] we have shown that for the source term of breakage and the sink term of coagulation, which are presented by linear and quasi-linear operators, it is possible to achieve a discretisation complexity of O ( n ) . For these two operators for further popular kernel functions the ideas of H -matrices were used. For the source term of coagulation, which is a quadratic operator, a more ambitious approach is necessary. We introduce a numerical treatment for two popular kernel functions combining the ideas of H -matrices and the fast Fourier transformation (FFT). This leads to almost linear complexity of O ( n log ⁡ n ) and O ( n log ⁡ 2 n ) but it turns unfortunately out that the new method is only applicable in a reasonable way for one of the two kernel functions.

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