The partial differential equation of groundwater movement mainly includes the first and second derivatives of the head distribution function. The finite difference method is based on the differential approximation of the derivative to establish the numerical model. Explained by one-dimensional problem, as shown in Figure 7. 1, the distribution of water head H along the X axis at a certain moment is a continuous function H(x), and the X axis is dispersed into nodes (x 1, x2, x3, …, xi, …, xN), and the corresponding water head values are (H 1, H2, H3) respectively.

Groundwater motion equation

Where Δ x = xi+1-xi is the node spacing, where o (Δ x) represents the function of error as Δ x, and the remainder of Taylor expansion in equation (7.5) is deleted, which is the first-order derivative forward difference scheme with first-order accuracy (Chen Chongxi et al., 1990). The first derivative can also be expressed as backward difference scheme, that is

Groundwater motion equation

And central difference scheme:

Groundwater motion equation

Based on the difference scheme of the first derivative and using the central difference scheme again, the difference scheme of the second derivative can be established:

Groundwater motion equation

Among them, the first derivative adopts backward difference scheme at Xi+ 1/2 and forward difference scheme at Xi- 1/2, and the difference scheme of the final second derivative is as follows.

Groundwater motion equation

If the distances between adjacent nodes are all equal, that is, Δ xi+1= Δ xi = Δ xi-1= Δ x, the equation (7.9) can be simplified as follows.

Groundwater motion equation

The difference schemes of the first and second derivatives are established, which can be used to establish the mathematical model of groundwater movement.