The generalized approximation method for modeling coagulation kinetics—Part 2: comparison with other methods

A detailed comparison of the generalized approximation (GA) method with grid and integral methods for modeling coagulation kinetics is performed. The method involves approximating the size distribution, C(g,t), by a sum of I delta functions, i.e. C(g,t)=∑i=1I Ci(t)δ[g−gi(t)]. Implicit differential equations for the 2I unknowns {Ci,gi} are obtained from evolution equations for the moments of the size distribution. Four different coagulation kernels are considered. In general, I=4 (i.e. solving eight differential equations) gives converged answers, of comparable accuracy to grid methods using 70 or more grid points. However, for the kernel Kc(g,n)=(g1/3+n1/3)g2/3−n2/3, the accuracy of all methods is poor at long times due to the existence of a “critical point” in time, at which the second moment of the size distribution becomes infinite. Comparison with other integral methods show that the GA method is more accurate, although somewhat more computationally intensive than methods employing Laguerre quadrature and associated Laguerre quadrature.