Possible Solution 1
Predictor–corrector method is an algorithm thatproceeds in two steps. First, the prediction step calculates a roughapproximation of the desired quantity. Second, the corrector step refines theinitial approximation using another means. A predictor**–corrector methodtypically uses an explicit method for the predictor step and animplicit method for the corrector step. A simple predictor–corrector method canbe constructed from the Eulermethod (an explicit method) and the trapezoidal rule (an implicit method).
Effectivenessand
Characteristics
The idea behind the predictor-corrector methods is to use asuitable combination of an explicit and an implicit technique to obtain amethod with better convergence characteristics Runge Kutta method Euler Method, Trapezoidal Rule arepredictor corrector methods.
Possible Solution 2
1. The P-C methods may provide both high accuracy and the capability of error monitoring,all at a potentially lower computational cost than RK-Fehlberg or RK-Merson methods.For example, the Adams-Moulton method (3.34) has the error of the same (fourth) orderas the aforementioned RK methods, while requiring k + 1 function evaluations, where kis the number of times one has to iterate the corrector equation. If k < 4, then Adam-Moulton requires fewer function evaluations than either RK-Merson or RK-Fehlberg
2. The adjustment of the step size in P-C methods is awkward (as it is in all multistepmethods); see the end of Sec. 3.1 and Remark 3 above.
3. One may ask, why not just halve the step size of the Adams-Bashforth method (whichwould reduce the global error by a factor of 24 = 16, i.e. a lot) and then use it alonewithout the Adams-Moulton corrector formula? The answer is this. First, one willthen lose the ability to monitor the error. Second, the Adams-Bashforth may sometimesproduce a numerical solution which has nothing to do with the exact solution, whilethe P-C Adams-Moulton's solution will stay close to the exact one
Predictor–corrector method is an algorithm thatproceeds in two steps. First, the prediction step calculates a roughapproximation of the desired quantity. Second, the corrector step refines theinitial approximation using another means. A predictor**–corrector methodtypically uses an explicit method for the predictor step and animplicit method for the corrector step. A simple predictor–corrector method canbe constructed from the Eulermethod (an explicit method) and the trapezoidal rule (an implicit method).
The idea behind the predictor-corrector methods is to use asuitable combination of an explicit and an implicit technique to obtain amethod with better convergence characteristics Runge Kutta method Euler Method, Trapezoidal Rule arepredictor corrector methods.
Possible Solution 2
1. The P-C methods may provide both high accuracy and the capability of error monitoring,all at a potentially lower computational cost than RK-Fehlberg or RK-Merson methods.For example, the Adams-Moulton method (3.34) has the error of the same (fourth) orderas the aforementioned RK methods, while requiring k + 1 function evaluations, where kis the number of times one has to iterate the corrector equation. If k < 4, then Adam-Moulton requires fewer function evaluations than either RK-Merson or RK-Fehlberg
2. The adjustment of the step size in P-C methods is awkward (as it is in all multistepmethods); see the end of Sec. 3.1 and Remark 3 above.
3. One may ask, why not just halve the step size of the Adams-Bashforth method (whichwould reduce the global error by a factor of 24 = 16, i.e. a lot) and then use it alonewithout the Adams-Moulton corrector formula? The answer is this. First, one willthen lose the ability to monitor the error. Second, the Adams-Bashforth may sometimesproduce a numerical solution which has nothing to do with the exact solution, whilethe P-C Adams-Moulton's solution will stay close to the exact one
