Elementary Finite Difference Quotients

Finite difference representations of derivatives are derived from Taylor series expansions. For example, if is the - component of the velocity, at point can be expressed in terms of Taylor series expansion about point as

...

(2.1)

Mathematically, Eq. (2.1) is an exact expression for if the series converges.

In practice, is small and any higher-order term of is smaller than . Hence, for any function Eq. (2.1) can be truncated after a finite number of terms.

Example:
In terms of magnitude, and higher order are neglected, Eq. (2.1) becomes

(2.2)

Eq. (2.2) is second-order accurate, because terms of order and higher have been neglected. If terms if order and higher are neglected, Eq. (2.2) is reduced to

(2.3)

Eq. (2.3) is first-order accurate.

In Eqns. (2.2) and (2.3) the neglected higher-order terms represent the truncation error. Therefore, the truncation errors for Eqns. (2.2) and (2.3) respectively are

and

It is now obvious that the truncation error can be reduced by retaining more terms in the Taylor series expansion of the corresponding derivative and reducing the magnitude of .