Backward differentiation formula

Remarks

It is an implicit linear multistep method in which only $\beta_k \ne 0$. Example: for $k = 1$ the formula is $y_{j+1} - y_j = h f(t_{j+1}, y_{j+1})$, the backward Euler method. For $k = 2$, $\nabla y_{j+1} + \frac12 \nabla^2 y_{j+1} = \frac32 y_{j+1} - 2 y_j + \frac12 y_{j-1}$, so the formula is $y_{j+1} - \frac43 y_j + \frac13 y_{j-1} = \frac23 h f(t_{j+1}, y_{j+1})$. Non-example: the Adams–Bashforth formula $y_{j+1} - y_j = h \left( \frac32 f_j - \frac12 f_{j-1} \right)$ is a linear multistep method but not a BDF, since it uses $f$ at old points and not at $t_{j+1}$. The $k$-step formula has order $k$; it is zero-stable for $1 \le k \le 6$ and not zero-stable for $k \ge 7$; the formulas with $k = 1, 2$ are A-stable.