The partial differential equation that describes steady-state flow in a two-dimensional, homogeneous, isotropic region of flow (Section 2.11) is Laplace’s equation:
To find the finite-difference equation for an interior node in the nodal grid used to discretize the region of flow, we must replace the second-order partial derivatives in Eq. (A6.1) by differences. Let us consider the first term of the equation first. Recall that the definition of the partial derivative with respect to of a function of two variables is
On a digital computer it is impossible to take the limit as , but it is possible to approximate the limit by assigning to some arbitrarily small value; in fact, we can do so by designing a nodal network with a mesh spacing of .
For any value of , say , we can expand in a Taylor’s expansion about the point as follows:
If we let (this is known as a forward difference) and abandon all the terms of order greater than unity, we can approximate by
The abandoned terms of the Taylor expansion represent the truncation error in the finite-difference approximation.
We can obtain a similar expression to Eq. (A6.4) by substituting the backward difference, , into Eq. (A6.3). This yields
To obtain the approximation for , we write the difference equation in terms of using a forward-difference expression:
and substitute the backward-difference expression Eq. (A6.5) in Eq (A6.6) to get
In a similar manner, we can develop a difference expression for as
For a square grid,; adding Eqs. (A6.7) and (A6.8) to for Laplace’s equation yields
If we let be the nodal point , Eq. (A6.9) can be rearranged to yield
which is identical to Eq. (5.24).