Implicit (Backward in Time) Finite Difference Method [for )]:
which can be written
Since Eq. 17 involves five unknown U-values at the time level , its solution requires that a system of linear equations be satisfied. The structure of these linear equations and some techniques for dealing with these equations are discussed in the next section. The implicit method Eq. 17 can be shown to provide a stable finite difference scheme for all choices of rx and ry.
We now turn our attention to the case that the coefficients c and are not constant. First, we consider the case that c and depend explicitly on x, y, and t. To facilitate this discussion, consider a grid labeled as indicated in Figure 3.
Figure 3: Block centered finite difference grid.