# Linear multistep methods for non-linear time steps

Consider an IVP $$dx/dt = f(x, t)$$, $$t \in [0, T]$$, $$f(x, 0) = f_0$$. An Adams-Bashforth method solves the IVP numerically by evaluating $$g_x(t_n) := f(x, t)$$ where $$t_n = h * n$$, $$n \in [0, T / h]$$, and importantly $$h$$ is constant.

How difficult would it be to adjust the Adams-Bashforth method, evaluating $$g_x$$ at arbitrary times $$\tilde t_0 < \dots < \tilde t_N$$. My motivation is that I can only evaluate $$f$$ for certain time values.

An example of this for the 2nd order Adams-Bashforth method is provided here: Using Adams-Bashforth-Moulton Predictor Corrector with Adaptive Step-size. Can this be "scaled" to arbitrary order methods?

• It seems to me that your only choice is to interpolate the values of $f$ that you need from the values that you can evaluate. Is this something that is feasible in your context? It is difficult to be more specific unless you add more details. In particular, is it true that you can evaluate $f(x,t)$ for arbitrary $x$, but only a very limited set of $t$ values? If the answer is yes, then there are some easy options. If the answer is no, then we need to be more sophisticated. The order of the interpolation should match the order of the time integrator. Feb 15 at 17:08
• Yes, I can evaluate $f(x, t)$ for arbitrary $x$. I realized, as you said, that you can simply use the Legendre polynomial also in this case, analogously to the post I linked. Feb 17 at 14:58