# Benchmark Systems for ODE Solvers - Reference Request

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.

• So you want more complicated systems than the example you presented ? – Piyush Grover Feb 20 '17 at 15:53
• @PiyushGrover They should be more complicated in the sense they reveal strengths and weaknesses of different ODE solvers. Another example may be 2nd order linear system with a large stiffness ratio and so on. I could take the time and try to enumerate them, but I am hoping to find a reference that already did something similar. This was already done in mentioned articles, but with local error as one of criteria instead of global error. – Slaven Glumac Feb 20 '17 at 17:28
• Focussing on systems with closed-form solutions pretty much excludes chaotic systems, which are ones where global error can be severe (butterfly effect etc.) – Robert Israel Feb 20 '17 at 17:56
• @Robert Israel I realize that this is the case and this fine for me. I would like to make experiments with such demonstration systems for introduction purposes. – Slaven Glumac Feb 20 '17 at 20:28
• You can easily come up with stiff systems which are analytically solvable. Basically, you need the eigenvalues to be of very different sizes, say $\lambda_1=10^3\lambda_2$. – Piyush Grover Feb 20 '17 at 22:13

It's easy to come up with a differential equation that has a known general solution, expressed in the form $F(t,y) = constant$ where $F$ is differentiable:
$$\dfrac{\partial F}{\partial t} + \dfrac{\partial F}{\partial y} y' = 0$$
$$F_i(t, y_1, \ldots, y_n) = c_i,\ i=1 \ldots,n$$ differentiation gives you $$\dfrac{\partial F_i}{\partial t} + \sum_{j=1}^n \dfrac{\partial F_i}{\partial y_j} y_j' = 0$$ which (when the Jacobian is invertible) you then solve for $y'_1, \ldots, y'_n$.