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.
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