4
Finite Difference Methods and the
Method of Characteristics for Solving
Hydrodynamic Dispersion Equations
4.1 Finite Difference Methods
4.1.1 Finite Difference Approximations of Derivatives
We have known the basic idea of finite difference methods (FDM) in the study
of groundwater flow problems. FDM includes three major steps. First, the
flow region is divided by a grid and the time interval into time steps. Second,
the partial derivatives involved in the PDE are replaced by their finite difference approximations. As a result, the PDE is transformed into a system of
algebraic equations. Third, the algebraic system is solved and the nodal
values of the unknown function are obtained. These discrete values approximately describe the time-space distribution of the unknown variable. We will
see that exactly the same steps can be used to solve advection-dispersion
problems.
Consider that there are three adjacent nodes along the x direction,
(x - Ax, y, z), (x, y, z) and (x + Ax, y, z) in the center of each cube, as shown in
Figure 4.1. If the nodes are numbered in accordance with the number of grid
lines, they can be denoted by (i - I,j, k), (i,j, k) and (i + I,j, k). The spatial
distances between the mesh planes in the three directions are assumed to be
Ax, Ay, and Az. The Taylor expansions centered around node (x, y, z, t) are
ac
a 2 C(Ax)2
3
C(x + Ax,y,z,t) = C(x,y,z,t) + ax Ax + ax 2 -2- + O[(Ax)] (4.1.1)
and
ac
a 2 C (AX)2
3
C(x - Ax,y,z,t) = C(x,y,z,t) - ax Ax + ax 2 -2- + O[(Ax)]. (4.1.2)
In these equations, the values of the partial derivatives on the right-hand
sides are all evaluated at point (x,y,z,t). From Eq. (4.1.1), we have
ac = C(x + Ax, y, z, t) - C(x, y, Z, t) O(A)
ax
Ax
+
x,
(4.1.3)
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