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.

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