1 Idea and Derivation of the Method
5
Table 1.1 Finite difference approximations for various differentiations, partly taken from [2, 4]
Derivative
Finite difference approximation
Type
Error
du
dX
i
u i+1 −u i
X
Forward diff. O((X )
−3u i +4u i+1 −u i+2
2X
”
O((X 2 )
u i −u i−1
X
Backward diff. O((X )
3u i −4u i−1 +u i−2
2X
”
O((X 2 )
u i+1 −u i−1
2X
Centered diff. O((X 2 )
d 2 u
dX 2
i
u i+2 −2u i+1 +u i
X 2
Forward diff. O((X )
−u i+3 +4u i+2 −5u i+1 +2u i
X 2
”
O((X 2 )
u i −2u i−1 +u i−2
X 2
Backward diff. O((X )
2u i −5u i−1 +4u i−2 −u i−3
X 2
”
O((X 2 )
u i+1 −2u i +u i−1
X 2
Centered diff. O((X 2 )
d 3 u
dX 3
i
u i+3 −3u i+2 +3u i+1 −u i
X 3
Forward diff. O((X )
−3u i+4 +14u i+3 −24u i+2 +18u i+1 −5u i
2X 3
”
O((X 2 )
u i −3u i−1 +3u i−2 −u i−3
X 3
Backward diff. O((X )
5u i −18u i−1 +24u i−2 −14u i−3 +3u i−4
2X 3
”
O((X 2 )
u i+2 −2u i+1 +2u i−1 −u i−2
2X 3
Centered diff. O((X 2 )
d 4 u
dX 4
i
u i+4 −4u i+3 +6u i+2 −4u i+1 +u i
X 4
Forward diff. O((X )
−2u i+5 +11u i+4 −24u i+3 +26u i+2 −14u i+1 +3u i
X 4
”
O((X 2 )
u i −4u i−1 +6u i−2 −4u i−3 +u i−4
X 4
Backward diff. O((X )
3u i −14u i−1 +26u i−2 −24u i−3 +11u i−4 −2u i−5
X 4
”
O((X 2 )
u i+2 −4u i+1 +6u i −4u i−1 +u i−2
X 4
Centered diff. O((X 2 )