Finite difference coefficient

In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward.

Contents

Central finite difference

This table contains the coefficients of the central differences, for several order of accuracy:

Derivative Accuracy −4 −3 −2 −1 0 1 2 3 4
1 2       −1/2 0 1/2      
4     1/12 −2/3 0 2/3 −1/12    
6   −1/60 3/20 −3/4 0 3/4 −3/20 1/60  
8 1/280 −4/105 1/5 −4/5 0 4/5 −1/5 4/105 −1/280
2 2       1 −2 1      
4     −1/12 4/3 −5/2 4/3 −1/12    
6   1/90 −3/20 3/2 −49/18 3/2 −3/20 1/90  
8 −1/560 8/315 −1/5 8/5 −205/72 8/5 −1/5 8/315 −1/560
3 2     −1/2 1 0 −1 1/2    
4   1/8 −1 13/8 0 −13/8 1 −1/8  
6 −7/240 3/10 −169/120 61/30 0 −61/30 169/120 −3/10 7/240
4 2     1 −4 6 −4 1    
4   −1/6 2 −13/2 28/3 −13/2 2 −1/6  
6 7/240 −2/5 169/60 −122/15 91/8 −122/15 169/60 −2/5 7/240

For example, the third derivative with a second-order accuracy is

\displaystyle f'''(x_{0}) \approx \displaystyle \frac{-\frac{1}{2}f(x_{-2}) %2B f(x_{-1}) -f(x_{%2B1}) %2B\frac{1}{2}f(x_{%2B2})}{h^3_x} %2B O\left(h_x^2 \right).

Forward and backward finite difference

This table contains the coefficients of the forward differences, for several order of accuracy:

Derivative Accuracy 0 1 2 3 4 5 6 7 8
1 1 −1 1              
2 −3/2 2 −1/2            
3 −11/6 3 −3/2 1/3          
4 −25/12 4 −3 4/3 −1/4        
5 −137/60 5 −5 10/3 −5/4 1/5      
6 −49/20 6 −15/2 20/3 −15/4 6/5 −1/6    
2 1 1 −2 1            
2 2 −5 4 −1          
3 35/12 −26/3 19/2 −14/3 11/12        
4 15/4 −77/6 107/6 −13 61/12 −5/6      
5 203/45 −87/5 117/4 −254/9 33/2 −27/5 137/180    
6 469/90 −223/10 879/20 −949/18 41 −201/10 1019/180 −7/10  
3 1 −1 3 −3 1          
2 −5/2 9 −12 7 −3/2        
3 −17/4 71/4 −59/2 49/2 −41/4 7/4      
4 −49/8 29 −461/8 62 −307/8 13 −15/8    
5 −967/120 638/15 −3929/40 389/3 −2545/24 268/5 −1849/120 29/15  
6 −801/80 349/6 −18353/120 2391/10 −1457/6 4891/30 −561/8 527/30 −469/240
4 1 1 −4 6 −4 1        
2 3 −14 26 −24 11 −2      
3 35/6 −31 137/2 −242/3 107/2 −19 17/6    
4 28/3 −111/2 142 −1219/6 176 −185/2 82/3 −7/2  
5 1069/80 −1316/15 15289/60 −2144/5 10993/24 −4772/15 2803/20 −536/15 967/240

For example, the first derivative with a third-order accuracy and the second derivative with a second-order accuracy are

\displaystyle f'(x_{0}) \approx \displaystyle \frac{-\frac{11}{6}f(x_{0}) %2B 3f(x_{%2B1}) -\frac{3}{2}f(x_{%2B2}) %2B\frac{1}{3}f(x_{%2B3}) }{h_{x}} %2B O\left(h_{x}^3 \right),
\displaystyle f''(x_{0}) \approx \displaystyle \frac{2f(x_{0}) - 5f(x_{%2B1}) %2B 4f(x_{%2B2}) - f(x_{%2B3}) }{h_{x}^2} %2B O\left(h_{x}^2 \right),

while the corresponding backward approximations are given by

\displaystyle f'(x_{0}) \approx \displaystyle \frac{\frac{11}{6}f(x_{0}) - 3f(x_{-1}) %2B\frac{3}{2}f(x_{-2}) -\frac{1}{3}f(x_{-3}) }{h_{x}} %2B O\left(h_{x}^3 \right),
\displaystyle f''(x_{0}) \approx \displaystyle \frac{2f(x_{0}) - 5f(x_{-1}) %2B 4f(x_{-2}) - f(x_{-3}) }{h_{x}^2} %2B O\left(h_{x}^2 \right).

In general, to get the coefficients of the backward approximations, give all odd derivatives listed in the table the opposite sign, whereas for even derivatives the signs stay the same.

See also

Finite difference methods
Finite volume methods
Finite element methods
Other methods
Domain decomposition methods

References

External links