Predictor–corrector method: Difference between revisions

Revision as of 23:05, 19 August 2012

Latest revision as of 17:19, 28 November 2024

(21 intermediate revisions by 13 users not shown)

Line 1:

{{Short description|Algorithms in numerical analysis}}

In [[mathematics]], particularly [[numerical analysis]], a '''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.

In [[numerical analysis]], '''predictor–corrector methods''' belong to a class of [[algorithm]]s designed to integrate ordinary differential equations{{snd}}to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps:

__TOC__

# The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new point.

# The next, "corrector" step refines the initial approximation by using the ''predicted'' value of the function and ''another method'' to interpolate that unknown function's value at the '''same''' subsequent point.

== 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 ===

A simple predictor–corrector method (known as [[Heun's method]]) can be constructed from the [[Euler method]] (an explicit method) and the [[trapezoidal rule (differential equations)|trapezoidal rule]] (an implicit method).

Consider the differential equation

Line 13 ⟶ 17:

: <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,

Line 19 ⟶ 23:

: <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>

Line 27 ⟶ 31:

=== 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}

Line 45 ⟶ 49:

: <math> \begin{align}

\tilde{y}_{i+1} &= y_i + h f(t_i,y_i), \\

\hat{y}_{i+1} &= y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\tilde{y}_{i+1}) \bigr)., \\

y_{i+1} &= y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\hat{y}_{i+1}) \bigr).

\end{align} </math>

The PECEC mode has one fewer function evaluation. than PECECE mode.

More generally, if the corrector is run ''k'' times, the method is in P(EC)^''k''

or P(EC)^''k''E mode. If the corrector method is iterated until it converges, this could be called PE(CE)^∞. <ref>{{harvnb|Butcher|2003|p=104}}</ref>

== See also ==

Line 69 ⟶ 75:

== External links ==

* {{MathWorld |title=Predictor-Corrector Methods |urlname=Predictor–Corrector MethodsPredictor-CorrectorMethods}}

* [https://web.archive.org/web/20080617035745/http://www.fisica.uniud.it/~ercolessi/md/md/node22.html Predictor–corrector methods] for differential equations

{{Numerical integrators}}

{{DEFAULTSORT:Predictor-corrector method}}

[[Category:Algorithms]]

[[Category:Numerical analysis]]

[[nl:Predictor-correctormethode]]

[[ru:Схема предиктор-корректор]]