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.