Das Adam-Bashforth Verfahren ist ein mehrstufiges Zeitintegrationsverfahren mit
$latex y(t_{n+1}) = y(t_n) + \Delta t \sum_{j=0}^m b_j f(t_{n-j},~y(t_{n-j}))$
Dabei sind $latex b_j$ die AB-Koeffizienten. Und $latex f(t, y(t)) = \frac{\partial y}{\partial t}(t,y(t))$. The AB-Coefficients can be calculated in the following way:
$latex b_j = \frac{(-1)^j}{j!(m-j)!} \int_0^1 \prod_{i=0,i\neq j}^{m} (v+i) \text{d}v$
Table of Adams-Bashforth-Coefficients up to order 10
The following table contains the resulting coefficients calculated with the formula mentioned above.
| $latex b_0$ | $latex b_1$ | $latex b_2$ | $latex b_3$ | $latex b_4$ | $latex b_5$ | $latex b_6$ | $latex b_7$ | $latex b_8$ | $latex b_9$ | $latex b_{10}$ | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0. order | 1 | ||||||||||
| 1. order | 1.5 | -0.5 | |||||||||
| 2. order | 1.9167 | -1.3333 | 0.4167 | ||||||||
| 3. order | 2.2917 | -2.4583 | 1.5417 | -0.375 | |||||||
| 4. order | 2.6403 | -3.8528 | 3.6333 | -1.7694 | 0.3486 | ||||||
| 5. order | 2.9701 | -5.5021 | 6.9319 | -5.0681 | 1.9979 | -0.3299 | |||||
| 6. order | 3.2857 | -7.3956 | 11.6658 | -11.3799 | 6.7318 | -2.2234 | 0.3156 | ||||
| 7. order | 3.59 | -9.5252 | 18.0545 | -22.0278 | 17.3797 | -8.6121 | 2.4452 | -0.3042 | |||
| 8. order | 3.8848 | -11.8842 | 26.3108 | -38.5404 | 38.0204 | -25.1247 | 10.7015 | -2.6632 | 0.2949 | ||
| 9. order | 4.1718 | -14.4669 | 36.642 | -62.6463 | 74.1793 | -61.2836 | 34.8074 | -12.9943 | 2.8776 | -0.287 | |
| 10. order | 4.452 | -17.2688 | 49.2505 | -96.2691 | 133.0191 | -131.8914 | 93.6472 | -46.617 | 15.4862 | -3.0889 | 0.2802 |
The table can be downloaded as an excel file: AB-Coefficients.