With
$$
\tilde{h}=h\left(\frac{\xi}{\epsilon}\right)^{1+1/M}.
$$
we can design the following algorithm:
- If the two answers are close, keep the approximation to \( h \).
- If \( \epsilon > \xi \) we need to decrease the step size in the next time step.
- If \( \epsilon < \xi \) we need to increase the step size in the next time step.