I would like to have a basic models as a ground truth for the numerical solvers. I am looking for systems which have available analytic solution. As an example I know that the closed form solution of IVP:
\begin{align}
 &\dot{y}(t) = -2y(t)\\
 &y(0) = 1
\end{align}
is:
\begin{align}
 y(t) = e^{-2t}
\end{align}
When using forward euler I know that numerical solution is given by the recurrence relation:
\begin{align}
 &y\_k = (1-2\Delta{t})y\_{k-1}\\
 &y\_0 = y(0)
\end{align}
This gives me a way of calculating **global error** for this specific system:
\begin{align}
e\_k = |y\_k - y(t\_k)|
\end{align}
The problem is that the above system may not be "relevant" as a benchmark for numerical solvers. I am looking for relevant ones as it is much easier to present the results in that case. There are articles on this topic such as
"Hull et al.: Comparing numerical methods for ordinary differential equations" and "Enright et al.: Comparing numerical methods for stiff systems of ODE:s". The problem is that I am looking for a way to present results using a global error. Is there any similar calculating closed form solutions?


Any suggestions and/or critiques are appreciated.



Benchmark Systems for ODE Solvers - Reference Request
There is a whole subfield of applied mathematics devoted to developing ODE solvers and understanding their properties. Consequently, there are thousands of relevant papers, and not much more can be said without a more specific question. The canonical reference, which includes lots of methods tested on lots of ODE systems (with global error as a metric) is


[Hairer, Norsett, Wanner. Solving ordinary differential equations (2 volumes)](http://www.unige.ch/~hairer/books.html).


