Predictor–corrector method: Difference between revisions

Revision as of 21:41, 8 February 2014

Revision as of 12:36, 6 January 2015

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== Predictor–corrector methods for solving ODEs ==

When considering the [[numerical methods for ordinary differential equations|numerical solution of ordinary differential equations (ODEs)]], a predictor­­–correctorpredictor–corrector method typically uses an [[explicit and implicit methods|explicit method]] for the predictor step and an implicit method for the corrector step.

=== Example: Euler method with the trapezoidal rule ===

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: <math> y' = f(t,y), \quad y(t_0) = y_0, </math>

and denote the step size by <math>h</math>.

First, the predictor step: starting from the current value <math>y_i</math>, calculate an initial guess value <math>\tilde{y}_{i+1}</math> via the Euler method,

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: <math>\tilde{y}_{i+1} = y_i + h f(t_i,y_i). </math>

Next, the corrector step: improve the initial guess using trapezoidal rule,

: <math> y_{i+1} = y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\tilde{y}_{i+1}) \bigr). </math>

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=== PEC mode and PECE mode ===

There are different variants of a predictor­–correctorpredictor–corrector method, depending on how often the corrector method is applied. The Predict–Evaluate–Correct–Evaluate (PECE) mode refers to the variant in the above example:

: <math> \begin{align}