Actual global error vs theoretical global error: How to combine theory with practice

Question

I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation:

$y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=e^{\lambda t}$. I have set $\lambda =1$.

I get decent results and decent rate of convergence because the loglog plot is linear with slope of 4 (AD4 is of order 4 so the error is $O(h^4)$). Here is the global error when I am testing it in the interval $t \in [0,3]$

Is this consistent with theory? Also, where will you plot the $O(h^4)-$line in this graph? I know it needs to be parallel to the AD4 line but where? In general, I don't really know the connection between the global error and $O(h^4)$

Here is my data

h        = ( 0.1, 0.05, 0.01, 0.005)
ad4error = (3.477e-03, 2.493e-04, 4.445e-07, 2.816e-08)  

Answer 1

I'm going to turn my comment into an answer.

The error order tells you up to which order (exclusive) the discrete solution corresponds to the exact solution.

An $\mathcal{O}(\Delta t^4)$ method reproduces the orders 0-3 of the Taylor expansion of the function around $\Delta t$ (i.e. in the expression $f(t+\Delta t) = f(t) +\dots$).

Put bluntly (and mathematically imprecisely) if you halve the size of the time step $\Delta t$ the deviation will be smaller by $2^4$. This, of course, assumes that the error is computed at the same final time. Or, if you want to use the log-log plot effectively you can say that by reducing the step size by one order of magnitude $\Delta t = 10^{-1} \rightarrow \Delta t = 10^{-2}$ the deviation from the analytic solution will be reduced by 4 orders of magnitude, i.e., instead of $10^{-2}$ it will be $10^{-6}$.

To visualize the growth of an error one could plot curves of different orders ($\Delta t^2, \Delta t^3$ etc.) and try and compare the result to these curves. This is very finnicky in a log-log plot and results in a rather confusing plot. A better way is to define a function that will draw a slope triangle with appropriately labelled sides close to the curve(s) you want to show the behaviour of.

I have used this in the analysis of the implementation of $\mathcal{P}_2,\mathcal{P}_3$ finite elements in my thesis, as can be seen in the following plot.

Note that this plot would be really confusing if I were to add an entire curve for each growth order. Admittedly, the slope triangles could have been placed better, but they do show the growth order (one power of $2$ of mesh width corresponds to $n$ powers of $10$ on the error axis).