Next: 4.1.6 Runge-Kutta Method
Up: 4.1 Initial Value Problems
Previous: 4.1.4 Implicit Methods
4.1.5 Predictor-Corrector Method
Explicit predictor / implicit corrector
PC method: a) EC ansatz: step function for ;
b) general predictor-corrector schemes: linear NGB extrapolation;
parabolic NGB extrapolation
Predictor step:
- Extrapolate the function , using an NGB polynomial, into
.
- Formally integrate the r.h.s. in :
Adams-Bashforth predictor:
- Truncate at some term to obtain the various predictors in the table.
Adams-Bashforth predictors
Evaluation step:
As soon as the predictor is available, insert it in :
Corrector step:
- Again back-interpolate the function , using NGB, but
now starting at .
- Formally re-integrate the r.h.s. in :
Adams-Moulton corrector:
Adams-Moulton correctors
Finally: evaluate
.
Stability of PC schemes:
Intermediate between the lousy explicit and the excellent implicit methods.
EXAMPLE:
2nd order PC + relaxation equation:
stable for
.
(The bare predictor would have
.)
Next: 4.1.6 Runge-Kutta Method
Up: 4.1 Initial Value Problems
Previous: 4.1.4 Implicit Methods
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001