Based on the solution of the stochastic Landau-Lifshitz-Gilbert equation discretized for a ferromagnetic chain subject to a uniform temperature gradient, we present a detailed numerical study of the spin dynamics with a particular focus on finite-size effects. We calculate and analyze the net longitudinal spin current for various temperature gradients, chain lengths, and external static magnetic fields. In addition, we model an interface formed by a nonuniformly magnetized finite-size ferromagnetic insulator and a normal metal and inspect the effects of enhanced Gilbert damping on the formation of the space-dependent spin current within the chain. One aim of this study is the inspection of the spin-Seebeck effect beyond the linear response regime. We find that within our model the microscopic mechanism of the spin-Seebeck current is the magnon accumulation effect quantified in terms of the exchange spin torque. According to our results, this effect drives the spin-Seebeck current even in the absence of a deviation between the magnon and phonon temperature profiles. The influence of the dipole-dipole interaction and domain formation on the spin current is exposed and discussed. Our theoretical findings are in line with the recently observed experimental results by Agrawal et al. [Phys. Rev. Lett. 109, 107204 (2012)].