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We consider the Abelian sandpile model (ASM) on the large square lattice witha single dissipative site (sink). Particles are added by one per unit time atrandom sites and the resulting density of particles is calculated as a functionof time. We observe different scenarios of evolution depending on the value ofinitial uniform density (height) $h_0=0,1,2,3$. During the first stage of theevolution, the density of particles increases linearly. Reaching a criticaldensity $\rho_c(h_0)$, the system changes its behavior sharply and relaxesexponentially to the stationary state of the ASM with $\rho_s=25/8$. We foundnumerically that $\rho_c(0)=\rho_s$ and $\rho_c(h_0>0) \neq \rho_s$. Ourobservations suggest that the equality $\rho_c=\rho_s$ holds for more generalinitial conditions with non-positive heights. In parallel with the ASM, weconsider the conservative fixed-energy Abelian sandpile model (FES). Theextensive Monte-Carlo simulations for $h_0=0,1,2,3$ have confirmed that in thelimit of large lattices $\rho_c(h_0)$ coincides with the threshold density$\rho_{th}(h_0)$ of FES. Therefore, $\rho_{th}(h_0)$ can be identified with$\rho_s$ if the FES starts its evolution with non-positive uniform height $h_0\leq 0$.

Numerical Study of the Correspondence Between the Dissipative and Fixed Energy Abelian Sandpile Models