A mathematical models for rainfall erosion of the soil steep slope is established

which consists of the control equation of overland flow in inter-rill of soil steep slope

the control equation of water flow in rills of variable groove width of soil steep slope

the continuation equation of sediment in inter-rill of soil steep slope

and the continuity equation of sediment in rills of variable groove width of soil steep slope. Through the probability distribution of turbulent flow

the soil erosion equation is determined. The expressions of sediment source/sink terms and rill height and width are determined. To solve the model

two methods are used in combination. The finite element method is applied to discretize the mathematical model that consist of hydrodynamic equation

the motion equation

and the sediment continuity equation in space; and the finite difference method solves the problem in time domain. In this way

we could obtain the numerical calculation formula and the specific solution to the model.