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

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)

• "I'm a self-learner so I don't read books." proceeds to post question on SE. Not making fun of you but I found this amusing! – David Ketcheson Mar 6 '19 at 6:36
• There is no substitute for books. Writing books requires a massive effort and forces the author to review, revise and extend her presentation of all subjects and how they interact. Order books through your public library – Carl Christian Mar 6 '19 at 8:31
• You want to use a slope triangle and to position it close to the curve representing our actual data. This way you will be able to show the order of growth of the error in a compact and easily comprehensible manner. – Nox Mar 6 '19 at 16:31

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).

• To be honest your excellent answer is way beyond what I am asking for. I don't have the theoritical knowledge to follow you 100%. The global error for a method of order $p$ is bounded such that $|E_N|\ \leq Ch^p$. I am basically asking how to find that $C$ constant and plot the line on the graph shown in my post – k.dkhk Mar 7 '19 at 16:34
• If you want an empirical constant just divide the computed error for a given step-size by the step-size and take the maximum of the results, e.g. $max_i\lbrace \frac{|E(h_i)|}{h_i}\rbrace$. To plot a simple line (I advise against that) is then to simply generate $y_i = C\cdot h_i$ with the constant you obtained and the step-sizes $h_i$ and then plot it against $h_i$. – Nox Mar 7 '19 at 16:45
• Let's I find $C$ the way you have described here. Then I use this method/solver on a NEW IV problem, let's say: $y'=y*sin(t)$ and I find values for $y$ for a given $h$. Is it hen fair to say that my error is bounded such that is is $Ch^4$ at the most? I might not have been clear with my question but that is basically what I try estimate with the $O(h^p)$ error as I described. Thank you very much for your help so far by the way – k.dkhk Mar 7 '19 at 20:07
• The constant is implicitly dependent upon the equation. For each problem you'll find a different constant. But once you've found a constant for a particular equation then the estimate will likely be true for that specific equation. – Nox Mar 11 '19 at 20:41