Table of Contents

Finite difference coefficients


How to calculate coefficients

In this example, I will calculate coefficients for DF4:

Use Taylor series:

So here:

Or in Matrix shape:

Here, we are looking for first derivative, so f_n^1. We only need to invert system to get coefficients. Trick is to move \Delta_x^k on right vector. Resulting matrix is then easy to solve. At the end, we have:

So for our derivative f_n^1:

With the same method, it is possible to get coefficients for all type of derivative, centered and uncentered.

First derivative:

Second derivative:

Coefficients

Central finite difference

This table contains the coefficients of the central differences, for several orders 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
5 2 -1/2 2 -5/2 0 5/2 -2 1/2

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

Backward can be obtained by inverting signs.

(source : http://en.wikipedia.org/wiki/Finite_difference_coefficient)