Integrators of several orders in time to study the evolution of an aerosol by coagulation

We have obtained some numerical methods to integrate the coagulation equation for an aerosol. They are semi-implicit and stable regarding the time integration step, which can be freely chosen (a very important matter in numerical solution of differential equations). The methods are of two types: extrapolative (based on a previously known first-order semi-implicit formula) and purely semi-implicit, both mass-conservative. The same methodology used here to develop these new methods can be applied to improve the well-known sectional ones. The extrapolative and the semi-implicit methods are really of the order we had deduced from their analysis. However, as the order of the method increases, for small time steps, the roundoff causes the error no longer to behave as expected. The extrapolative methods are self-starting but the semi-implicit ones are not, so we need the first ones to start the others. If we take into account both the error and the CPU time, the second-order methods are comparable, but the third-order semi-implicit one is better than the extrapolative one. The comparison of higher order methods is disturbed by the roundoff error. Both methods can be used with fixed and moving bins with respect to the discretization of the size in the particle size distribution. These methods are valid to complement the specific ones developed to solve the growth and other phenomena in the time-splitting method which is used to analyse the evolution of an aerosol in the general case.