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In growth of nanosized clusters and dots on surfaces there are interesting cases, where size-selection of dots is associated with the kinetically determined metastable affected by both the energetics and kinetics of the growth. The detailed studies of the kinetics of the growth in these cases, however, have been hampered by the computational difficulties due to slow convergence towards the metastable state. In this work, we examine the size selected growth of nanodots by using a mesoscopic reaction kinetic model (RKM) with self-consistent reaction rates for size dependent attachment and detachment processes. The energetics of the adatom process is described through the free energy difference of the growing dots, which in the continuum limit is equal to the chemical potential. We introduce here two effective computational schemes for tackling such problems. The first method is based on the particle coalescence method (PCM), where the configuration space of dots is sampled by using the rejection-free Bortz-Kalos-Lebowitz algorithm. The second method is based on direct numerical integration of the RKM by using the transformation referred as the Master Equation Discretization (MED) scheme. The MED scheme enables the use of spatial grid scales between two and four times larger than any traditional discretization schemes and thus allows for a better numerical efficiency without the loss of accuracy. We compare the computational efficiency of the PCM and RKM discretization methods in a typical case of 2D-nanocluster growth with size dependent energetics and show that both of these approaches allows us to study in detail the evolution of the size distribution in all stages of the growth from the initial stage, with high density of small dots to the final long-lived stationary state. Both methods are shown to be capable to describe effectively and with ease these computationally challenging stages of growth. |