Explicit (Forward in Time) Finite Difference Method [for ]:
where , , , and denote values of the coefficient to be used in calculating the flux across the west, east, south, and north faces of the control cell whose center is . The strategies for choosing the various -values on the faces of the control cell are similar to the single space variable case. If , one could set
In the case that , if the harmonic averages were chosen for calculating face -values, then we would set
Having assigned four face values of for each finite difference cell, we can express the explicit FDE for as
For stability of Eq. 29, the coefficient of must be nonnegative. In practice, it is often convenient to express equations such as Eq. 29 using a computational stencil as indicated in Figure 5. Note that the explicit FDE can be implemented by overlaying the stencil of Figure 5 on the array and forming the linear combination found on the right side of Eq. 29.
Figure 5: Computational stencil for the Explicit Method.
The development of the backward in time implicit FDE and the Crank--Nicolson method for the two dimensional, variable coefficient case is straightforward, and is left to the exercises. The extension of these ideas to three dimensions is also left as an exercise. In the following section we discuss some computational issues and techniques for solving the conservation law FDEs developed in this section.