A comparison of some approximate methods for solving the aerosol general dynamic equation

The general dynamic equation for aerosol evolution is converted into a set of differential equations for the rate of change of the moments, Mm by multiplying the equation by vm and integrating over all v. The integrals over the size distribution appearing in these equations are approximated by quadrature sums over the distribution evaluated at discrete quadrature points. The results from different quadrature schemes are compared with accurate numerical values from a finite element method for a variety of coagulation mechanisms with and without gravitational removal. In general, three-point Laguerre or associated Laguerre quadrature, involving the numerical solution of three differential equations, produces results of comparable accuracy to more complicated schemes. The method is very easy to implement and typically produces values within a few percent of the accurate results. Increasing the number of quadrature points, or the number of differential equations, does not significantly improve the accuracy. For gravitational coagulation (both with and without gravitational removal) all quadrature schemes produce inaccurate results or fail completely.