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) alpha (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 less than or equal to 1 and lambda = mu + nu less than or equal to 1, the numerical solution evolves in an orderly fashion and tends towards 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 behaviour at small mass, but in a damped oscillatory fashion that was not known previously. For nu < 1 and. > 1, the numerical solution tends towards a self-similar solution as t approaches a finite time to. The mass spectrum nk develops at t(0) a power-law tail n(k) alpha k(-tau) at large masses 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) --> infinity. 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 greater than or similar to 1.1. For nu > 1, the behaviours 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) --> infinity. They also indicate that the time t(crit) for the onset of runaway growth decreases slowly towards zero with increasing n(0).