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 ∝ ( i μ j ν + i ν j μ ) and monodisperse initial conditions. The results confirm that there are three classes of rate coefficients with qualitatively different solutions. For ν ≤ 1 and λ = μ + ν ≤ 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 μ > 0 and λ < 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 ν ≤ 1 and λ > 1, the numerical solution tends towards 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 ∝ k −τ 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 → ∞ . The exponent τ is in general less than the analytic prediction (λ + 3)/2, and t 0 = K / [(λ − 1) n 0 A 11 ] with K = 1–2 if λ ≳ 1.1. For ν > 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 → ∞ . They also indicate that the time t crit for the onset of runaway growth decreases slowly towards zero with increasing n 0 .