Predictor-corrector pairs

A compromise between the explicit and implicit methods is the predictor-corrector technique, where the explicit method is used to obtain a first estimate of and this estimated y +i is then used in the RHS of the implicit formula. The result is the corrected y + i, which should be a better estimate to the true y +i than the first estimate. The corrector formula may be applied several times (i.e., successive substitution) until the convergence criterion (7.49) is achieved. Generally, predictor-corrector pairs are chosen such that they have truncation errors of approximately the same degree in h but with a difference in sign, so that the truncation errors compensate one another. 

The implicit multistep methods add stability but require more computation to evaluate the implicit part. In addition, the error coefficient of the Adams-Moulton method of order k is smaller than that of the Adams Bashforth method of the same order. As a consequence, the implicit methods should give improved accuracy. In fact, the error coefficient for the imphcit fourth-order Adams Moulton method is 19/720, and for the explicit fourth-order Adams Bashforth method it is 251/720. The difference is thus about an order of magnitude. Pairs of exphcit and implicit multistep methods of the same order are therefore often used as predictor-corrector pairs. In this case, the explicit method is used to calculate the solution,, at v +i. Furthermore, the imphcit method (corrector) uses y + to calculate /(x +i,y +i), which replaces /(x +i,y +i). This allows the solution, y +i, to be improved using the implicit method. The combination of the Adams Bashforth and the Adams Moulton methods as predictor orrector pairs is implemented in some ODE solvers. The Matlab odel 13 solver is an example of a variable-order Adams Bashforth Moulton multistep solver. 

How do predictor corrector pairs work, and why are they nsed ... 

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