Elementary Finite Difference Quotients

Let us return to Eq. (2.1) and solve for as:

or

(2.4)

In Eq. (2.4) the symbol is a formal mathematical nomenclature which means “terms of order of ” , expressing the order of the magnitude of the truncation error. The first-order-accurate difference representation for the derivative expressed by Eq. (2.4) can be identified as a first-order forward difference.

Now consider a Taylor series expansion for , and

or

(2.5)

Solving for , we obtain

(2.6)

Eq. (2.6) is a first-order backward expression for the derivative at grid point

Subtracting Eq. (2.5) from (2.1)

(2.7)

And solving for from Eq. (2.7) we obtain

(2.8)

Eq. (2.8) is a second-order central difference for the derivative at grid point