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Finite Difference
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)
Poisson Equation
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Solving Poisson equation
Good way is to combine Conjugate Gradient and derived methods with Multi-grids methods. CG will take care of high frequencies while MG will concentrate on low frequencies. Both methods are know to be parallel (MPI), and with some tricks it is possible to make them scale.
Example used here will be 2D/3D heat equations for infinite time (looking for solution when system is stable), which is an easy way to test methods and to teach Poisson equation.
Conjugate Gradient
BI-Conjugate Gradient Stabilized
Red-Black Gauss Seidel Multi-grids
Algebraic Multi-grids
Make it scale !
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