Deformations and Liftings of Finite, Commutative Group Schemes.
Taylor Series expansion and Finite difference schemes
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![Page 1: Taylor Series expansion and Finite difference schemes](https://reader038.fdocuments.us/reader038/viewer/2022100420/55721201497959fc0b8fdfe3/html5/thumbnails/1.jpg)
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)
![Page 2: Taylor Series expansion and Finite difference schemes](https://reader038.fdocuments.us/reader038/viewer/2022100420/55721201497959fc0b8fdfe3/html5/thumbnails/2.jpg)
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.
![Page 3: Taylor Series expansion and Finite difference schemes](https://reader038.fdocuments.us/reader038/viewer/2022100420/55721201497959fc0b8fdfe3/html5/thumbnails/3.jpg)
4- Order of Transport Equation:
( ) ( )
( ) ( )
( ) ( )
Divide by
( ) ( )
1st order scheme.
( ) ( )
( ) ( )
( ) ( )
Dividing by
( ) ( )
1st order scheme.