calculating the order of approximation in the Discretization scheme

Introduction:

In this topic we will discuss on how to find out the order of approximation of your discretization scheme. The order of approximation can be determined by the truncation error term on the RHS.

Mathematical Example:

let us take this exampe 2nd order derivative for example.

Let us try to derive a 1st order approximation for the above 2nd order derivative.

Using the Taylor\'s series exapansion

Using the same method you can also achieve the expansion of f(i+2), on further simplifying the equations we get

Now using the above equation we can come up with 3 linear equations

Since this is for a second derivative we have the co-efficients equal to 1 in the 3rd equations. on solving the 3 equations we get

a = 1

b = -2

c = 1

Now we can re-write the equations again as

This equations is approximately equal to

The term from the RHS can be written as the order term (truncation error)

Replacing this in the above equation we get

from the above equations we can also find out that

Dividing the above equation by del(x)^2 we get our result for 1st order skewed right hand side scheme.