1- Proof that finite difference central scheme for second derivative of a function is second

order scheme:

( ) ( ) ( )

( ) ( )

(1)

( ) ( )

(2)

Adding 2 to 1:

( ) ( ) ( )

( ) ( ) ( )

Dividing by

( ) ( ) ( )

2nd order scheme.

2- Proof that finite difference backward scheme for first derivative of a function is second

order scheme:

( ) ( ) ( )

( ) ( )

(1)

( ) ( )

(2)

Multiplying (1) by 4

( ) ( )

(3)

Subtracting (3) from (2):

( ) ( ) ( )

( ) ( ) ( )

Dividing by

( ) ( ) ( )

2nd Order scheme.

3- Proof that finite difference forward scheme for first derivative of a function is second

order scheme:

( ) ( ) ( )

( ) ( )

(1)

( ) ( )

(2)

Multiplying (1) by 4:

( ) ( )

(3)

Subtracting (2) from (3):

( ) ( ) ( )

( ) ( ) ( )

Dividing by

( ) ( ) ( )

2nd order scheme.

4- Order of Transport Equation:

( ) ( )

( ) ( )

( ) ( )

Divide by

( ) ( )

1st order scheme.

( ) ( )

( ) ( )

( ) ( )

Dividing by

( ) ( )

1st order scheme.