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A Survey of Numerical Solutions to the Coagulation Equation

We present the results of a systematic survey of numerical solutions to the coagulation equation for a rate coefficient of the form A_ij \propto (i^mu j^nu + i^nu j^mu) and monodisperse initial conditions. The results confirm that there are three classes of rate coefficients with qualitatively different solutions. For nu \leq 1 and lambda = mu + nu \leq 1, the numerical solution evolves in an orderly fashion and tends toward a self-similar solution at large time t. The properties of the numerical solution in the scaling limit agree with the analytic predictions of van Dongen and Ernst. In particular, for the subset with mu > 0 and lambda < 1, we disagree with Krivitsky and find that the scaling function approaches the analytically predicted power-law behavior at small mass, but in a damped oscillatory fashion that was not known previously. For nu \leq 1 and lambda > 1, the numerical solution tends toward a self-similar solution as t approaches a finite time t_0. The mass spectrum n_k develops at t_0 a power-law tail n_k \propto k^{-tau} at large mass that violates mass conservation, and runaway growth/gelation is expected to start at t_crit = t_0 in the limit the initial number of particles n_0 -> \infty. The exponent tau is in general less than the analytic prediction (lambda + 3)/2, and t_0 = K/[(lambda - 1) n_0 A_11] with K = 1--2 if lambda > 1.1. For nu > 1, the behaviors of the numerical solution are similar to those found in a previous paper by us. They strongly suggest that there are no self-consistent solutions at any time and that runaway growth is instantaneous in the limit n_0 -> \infty. They also indicate that the time t_crit for the onset of runaway growth decreases slowly toward zero with increasing n_0.