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.

  • $\begingroup$ So you want more complicated systems than the example you presented ? $\endgroup$ Feb 20, 2017 at 15:53
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    $\begingroup$ @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. $\endgroup$ Feb 20, 2017 at 17:28
  • $\begingroup$ Focussing on systems with closed-form solutions pretty much excludes chaotic systems, which are ones where global error can be severe (butterfly effect etc.) $\endgroup$ Feb 20, 2017 at 17:56
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    $\begingroup$ @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. $\endgroup$ Feb 20, 2017 at 20:28
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    $\begingroup$ You can easily come up with stiff systems which are analytically solvable. Basically, you need the eigenvalues to be of very different sizes, say $\lambda_1=10^3\lambda_2$. $\endgroup$ Feb 20, 2017 at 22:13

2 Answers 2


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).


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:

$$ \dfrac{\partial F}{\partial t} + \dfrac{\partial F}{\partial y} y' = 0 $$

Similarly, for a system, expressed in implicit form as

$$ F_i(t, y_1, \ldots, y_n) = c_i,\ i=1 \ldots,n $$ 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 $y'_1, \ldots, y'_n$.


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