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 t₀. The mass spectrum n_k develops at t₀ 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 = t₀ in the limit the initial number of particles n₀ → ∞. The exponent τ is in general less than the analytic prediction (λ + 3)/2, and t₀ = K / [(λ − 1)n₀A₁₁] 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 n₀ → ∞. They also indicate that the time t_crit for the onset of runaway growth decreases slowly towards zero with increasing n₀.