I'm investigating the behaviour of predictor-corrector methods to numerically give approximations to the Initial Value Problem.
I have currently implemented a Forward-Backward euler method like so,
$$\text{Predictor:}\quad u_{i+1}^*=u_i+h_if(u_i, t_i),\\ \text{Corrector:}\quad u_{i+1} = u_i + h_if(u_{i+1}^*,t_{i+1}),$$
with a variable step size $h_i$ which is calculated by the formula,
$$h_{i+1}=h_i \left \lvert \frac{\text{tolerance}}{T_{i+1}}\right\rvert ^{(p+1)^{-1}},$$
where $p$ is the order of the method (in this case $p=1$) and $T_{i+1}$ is the half the difference of the predictor and corrector (approximation for LTE).
My question comes when wanting to use 2,3-step Adams Bashforth-Moulton predictor-corrector combinations. I know how to calculate $p$, but I don't understand how, if I am changing the step-size at each step, I use the iterative schemes correctly. For example, take the Adams Bashforth 2 method for our predictor, with fixed step size
$$\text{Predictor}:\quad u_{i+1}=u_i+\frac{h}{2}\left ( 3f(u_i, t_i) - f(u_{i-1}, t_{i-1}) \right ).$$
What happens if the step size is adapted between step $i$ and step $i+1$. For clarity, let $h_i=t_{i+1}-t_i$. Do I need to factor in each step's length into our scheme (1)? Or Do I need to recompute previous intermediat u and derivative values using some kind of Runge-Kutta(2)?
$$(1):\quad u_{i+1}=u_i+\frac{3h_i}{2}f(u_i,t_i) - \frac{h_{i-1}}{2}f(u_{i-1},t_{i-1})$$