4.1.5 Predictor-Corrector Method

Explicit predictor / implicit corrector

PC method: a) EC ansatz: step function for $f(t)$; b) general predictor-corrector schemes: $1\dots$ linear NGB extrapolation; $2\dots$ parabolic NGB extrapolation

Predictor step:

- Extrapolate the function $f(t)$, using an NGB polynomial, into $[t_{n}, t_{n+1}]$.

- Formally integrate the r.h.s. in $dy/dt=f(t)$:
$\Longrightarrow$Adams-Bashforth predictor:

$$y^{P}_{n+1} = y_{n}+\Delta t \left[ f_{n} + \frac{1}{2} \nabla f_{n} + \frac{5}{12} \nabla^{2} f_{n}+ \frac{3}{8} \nabla^{3} f_{n} + \dots \right]$$

- Truncate at some term to obtain the various predictors in the table.



Adams-Bashforth predictors



Evaluation step:

As soon as the predictor $y^{P}_{n+1}$ is available, insert it in $f(y)$:

$$f_{n+1}^{P} \equiv f[y_{n+1}^{P}]$$

Corrector step:

- Again back-interpolate the function $f(t)$, using NGB, but now starting at $t_{n+1}$.

- Formally re-integrate the r.h.s. in $dy/dt=f(t)$:
$\Longrightarrow$Adams-Moulton corrector:

$$y_{n+1} = y_{n}+\Delta t \left[ f_{n+1} - \frac{1}{2} \nabla f_{n+1} - \frac{1}{12} \nabla^{2} f_{n+1} - \frac{1}{24} \nabla^{3} f_{n+1} - \dots \right]$$

Adams-Moulton correctors

Finally: evaluate $f_{n+1} \equiv f(y_{n+1})$.

Stability of PC schemes:

Intermediate between the lousy explicit and the excellent implicit methods.

EXAMPLE: 2nd order PC + relaxation equation: stable for $\Delta t \leq 2/\lambda$. (The bare predictor would have $\Delta t \leq 1/\lambda$.)