MTH603 GDB Idea Solution Spring July 2012

The topic for Graded Moderated Discussion Board is

“Compare the efficiency and characteristics of Predictor- Corrector methods available to you.”

---

  Solution:-
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) order as the aforementioned RK methods, while requiring k + 1 function evaluations, where k is 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 multi step methods); 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 (which would reduce the global error by a factor of 24 = 16, i.e. a lot) and then use it alone without the Adams-Moulton corrector formula? The answer is this. First, one will then lose the ability to monitor the error. Second, the Adams-Bashforth may sometimes produce a numerical solution which has nothing to do with the exact solution, while the P-C Adams-Moulton's solution will stay close to the exact one.





2nd Idea Solution:
Predictor–corrector method is an algorithm that proceeds in two steps. First, the prediction step calculates a rough approximation of the desired quantity. Second, the corrector step refines the initial approximation using another means. A predictor–corrector method typically uses an explicit method for the predictor step and an implicit method for the corrector step. A simple predictor–corrector method can be constructed from the Euler method (an explicit method) and the trapezoidal rule (an implicit method). Effectiveness and Characteristics

The idea behind the predictor-corrector methods is to use a suitable combination of an explicit and an implicit technique to obtain a method with better convergence characteristics Runge Kutta method Euler Method, Trapezoidal Rule are predictor corrector methods