Numerical analysis Ordinary differential equations Proposition

Predictor–corrector method

Builds on Linear multistep method Initial value problem

Statement

Let $f \colon \R \times \R^d \to \R^d$ be sufficiently smooth, and consider the initial value problem $y' = f(t, y)$ , $y(t_0) = y_0$ . Let $P$ be an explicit linear multistep method of order $p^* \ge 1$ , the predictor. Let $C$ be an implicit linear multistep method of order $p \ge 1$ , the corrector. One step of the PECE mode with step size $h$ does the following:

P: compute a predicted value $y_{n+1}^{[0]}$ with $P$ .
E: evaluate $f_{n+1}^{[0]} = f(t_{n+1}, y_{n+1}^{[0]})$ .
C: compute $y_{n+1}$ with the formula of $C$ , where $f_{n+1}^{[0]}$ replaces the unknown value $f(t_{n+1}, y_{n+1})$ .
E: evaluate $f_{n+1} = f(t_{n+1}, y_{n+1})$ for use in later steps.

Then the PECE method is an explicit method of order $\min(p, p^* + 1)$ . If $p^* \ge p$ , its principal local truncation error equals that of the corrector.

Remarks

The corrector is applied once instead of being iterated to convergence, so no nonlinear equation is solved. The error of the predicted value is $O(h^{p^*+1})$ . It enters the corrected value multiplied by $h$ times a coefficient of the corrector, so it adds $O(h^{p^*+2})$ to the local error. Comparing this with the local error $O(h^{p+1})$ of the corrector gives the order $\min(p, p^*+1)$ . Example: the Euler method ( $p^* = 1$ ) as predictor and the trapezoidal rule ( $p = 2$ ) as corrector give Heun's method, of order $\min(2, 2) = 2$ . When $p^* = p$ , the difference $y_{n+1} - y_{n+1}^{[0]}$ gives an estimate of the local error (Milne's device), which is used for adaptive step size control.

Going deeper

For a student

The Adams–Bashforth–Moulton pairs are the standard example: an Adams–Bashforth predictor with an Adams–Moulton corrector of the same order. The PECE method has the same order as the corrector, but its stability region is different and is in general smaller than that of the corrector. So predictor-corrector methods are not suited to stiff equations, where one solves the implicit equations with Newton's method instead.

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Tag	`MHGV8HCAQR`
URI	`m2://MHGV8HCAQR`
Kind	statement, proposition
Writing	W0 stub
MSC	65L06, 65L05
Status	This copy of the site has no build data. Run `m2 status MHGV8HCAQR`.
About	Linear multistep method
Uses	Linear multistep method, Initial value problem