The objective of this paper is to take into account the uncertainties in the rainfall-runoff process. To this end, this paper used a stochastic approach which is derived from the deterministic hydrological model based on the least action principle (ModHyPMA). The stochastic formulation of ModHyPMA allows to take into account both the dynamics and the stochastic character of the hydrological phenomenon. The main assumption is that the uncertainties in the hydrological process are modelled as Gaussian white noise. Moreover, we also assumed that hydrological systems are nonlinear dynamical systems that can be described by stochastic differential equations (SDE). From this SDE, we have deduced the associated Fokker-Planck equation (FPE). The FPE is a partial differential equation that cannot be solved analytically due to the complexity of its terms. We therefore, investigated a numerical solution to this equation by using the finite differences and finite volumes methods. The results shows that the stochastic model allows to improve the simulations of discharges in the Ouémé at Savè basin (NSE = 0.89, R2 = 0.90, RMSE = 113 and MAE = 76) in comparison to the deterministic model (NSE = 0.78, R2 = 0.78, RMSE = 123 and MAE = 51). From the comparison of the investigated numerical solutions, we noticed that the plots of the solutions (i.e. the density probability of discharges) are always coincided, except in the case of a very small number of meshes (100 meshes). We also found that the two solutions are convergent. This numerical solution gives us several pieces of information about the distribution of the discharges in the Ouémé at Savè basin.
Keywords: Uncertainty, stochastic approach, Fokker-Planck equation (FPE), ModHyPMA, numerical solutions, density probability.