Adaptive step size

Builds on Numerical methods for ordinary differential equations Local truncation error

Definition

Let a one-step method for $y' = f(t, y)$ come with an error estimator. From $t_n$ , $y_n$ and a trial step $h$ it computes $y_{n+1}$ and a number $\mathrm{err}_n \ge 0$ . This number estimates the norm of the local truncation error of the step, which behaves like $C h^{q+1}$ for some integer $q \ge 1$ . Let $\mathrm{tol} > 0$ be a tolerance, $0 < \rho < 1$ a safety factor, and $0 < \alpha_{\min} < 1 < \alpha_{\max}$ bounds. Adaptive step size control runs the method with steps chosen as follows.

If $\mathrm{err}_n \le \mathrm{tol}$ , the step is accepted: $t_{n+1} = t_n + h$ , and $y_{n+1}$ is kept.
If $\mathrm{err}_n > \mathrm{tol}$ , the step is rejected and repeated from $t_n$ .
In both cases the next trial step is $h_{\mathrm{new}} = h \cdot \min\Big(\alpha_{\max}, \max\Big(\alpha_{\min}, \; \rho \, \Big(\frac{\mathrm{tol}}{\mathrm{err}_n}\Big)^{1/(q+1)}\Big)\Big).$

Remarks

The formula of clause 3 comes from $\mathrm{err} \approx C h^{q+1}$ : the step $h'$ with $C h'^{q+1} = \mathrm{tol}$ is $h (\mathrm{tol}/\mathrm{err})^{1/(q+1)}$ , and $\rho$ , $\alpha_{\min}$ , $\alpha_{\max}$ protect against a poor estimate. Example: with an embedded pair of orders $4$ and $5$ such as the Runge–Kutta–Fehlberg method, $\mathrm{err}_n = \norm{\hat y_{n+1} - y_{n+1}}$ and $q = 4$ ; if $\mathrm{err}_n = 32\,\mathrm{tol}$ the step is rejected and, with $\rho = 0.9$ , retried with $h_{\mathrm{new}} = 0.9 \cdot 32^{-1/5} h = 0.45 h$ . Other estimators use step doubling: compare one step of size $h$ with two steps of size $h/2$ , as in Richardson extrapolation. Non-example: a schedule fixed in advance, such as $h_n = 2^{-n}$ , varies the step but is not adaptive, since it ignores the error estimate. The tolerance controls the local error per step; the global error is usually of a similar size but is not guaranteed to be below $\mathrm{tol}$ .

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Local truncation error concept
Numerical methods for ordinary differential equations concept
Adaptive step size definition

Tag	`F7H4H711MY`
URI	`m2://F7H4H711MY`
Kind	definition
Writing	W0 stub
MSC	65L50, 65L05, 65L06
Defines	Adaptive step size
Uses	Numerical methods for ordinary differential equations, Local truncation error

Adaptive step size

Definition

Remarks

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