Adaptive Stepsize Control

In order to assess the accuracy of numerical integration, and possibly adjust the stepsize so as to maintain the requested accuracy employ step doubling:

1.

take the step twice

(a)

once as a full step, leading to $x_1(t + \Delta t)$

(b)

then as two half steps, leading to $x_2(t + \Delta t)$

2.

Estimate the truncation error by

$$\Delta x (\Delta t) = x_2(t + \Delta t) - x_1(t + \Delta t) \approx {\cal O}\left( (\Delta t)^5 \right)$$

3.

Return $x_2(t + \Delta t)$ as an answer, because that's going to be the more accurate one.

4.

Since $\Delta x \approx {\cal O}\left( (\Delta t)^5 \right)$ assuming that we try two different values of $\Delta t$ we should have:

$$\frac{(\Delta x)_0}{(\Delta x)_1} = \left( \frac{(\Delta t)_0}{(\Delta t)_1} \right)^5$$

This yields the following formula for a step size:

$$(\Delta t)_0 = (\Delta t)_1 \left\vert\frac{(\Delta x)_0}{(\Delta x)_1}\right\vert^{1/5}$$ (4.78)

Strategy

Let $(\Delta x)_0$ be the requested accuracy.

• If $(\Delta x)_1 > (\Delta x)_0$ equation (4.84) tells us how much to reduce the stepsize when we repeat the failed step.
• If $(\Delta x)_1 < (\Delta x)_0$ equation (4.84) tells us how much we can stretch the stepsize for the next step.