Unformatted text preview: evaluations and, thus, important to select an accurate initial guess yn . Initial guess are
usually chosen by a separate explicit method called a predictor. The implicit method is
then called a corrector. For example, the Adams-Bashforth methods furnish predictors
for use with Adams-Moulton correctors.
Two possible termination criteria for the corrector iteration are: 46 1. Iterate the corrector to convergence, e.g., terminate when
jyn ) ; yn ;1) j where is on the order of the unit round o error of the computer. The local
discretization and stability of this predictor-corrector combination are determined
by the properties of the corrector alone however, more function evaluations than
necessary may be required.
2. Iterate the corrector a xed number of times. This procedure reduces the number of
function evaluations, but now the discretization error and stability characteristics
of the result contain a mixture of the properties of both the predictor and corrector.
A compromise between these two strategies is to iterate the corrector a xed number
of times and to repeat the step with a smaller step size if adequate convergence was not
In describing predictor-corrector methods, we will use the notation P to denote an application of the predictor step,
C to denote an application of the corrector step, and
E to denote a function evaluation, i.e., an evaluation of f .
Example 5.7.1. A method that uses a predictor to evaluate yn , does a function
evaluation fn = f (tn yn ), and one corrector iteration to obtain yn = yn is a PEC
method. If corrector iterations are performed and each iteration requires one evaluation
of f , then the method is a P (EC ) method. In this case, the last function evaluation
f (tn yn ;1)) is saved as fn for the next time step. Since we save yn = yn ), it may be
preferable to make an additional function evaluation and save fn = f (tn yn )). This
method then becomes a P (EC ) E method.
The orders of the predictor and corrector formulas need not be the same. Assume
that the predictor has an order p and the corrector has an order...
View Full Document