# Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous?

Existing algorithms for solving ODEs handle functions $\frac{dy}{dt} = f(y, t)$, where $y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so $\frac{dy}{dt} = f(y)$, $y \in \mathbb R^n$, with the $t$ left out. With this simplifying assumption, what improvements can be seen in existing numerical methods? For example, if $n=1$, the problem turns into $t = \int \frac{dy}{f(y)}$ and we turn to a wholly different class of algorithms for integrating one-dimensional integrals. For $n>1$, the maximum possible improvement is reducing the dimension of $y$ by 1, because the time-dependent case can be simulated by appending $t$ to $y$, changing the domain of $y$ from $\mathbb R^n$ to $\mathbb R^{n+1}$.

## 1 Answer

I would say one significant improvement is that in the scope of time-stepping approaches, where you propagate $$y_n \rightarrow y_{n+1}=\mathcal{U}(y_n)$$ using a solution map $$\mathcal{U}$$, you can determine the propagator (or at least parts of it) once and then re-use it at every time step.

For example, in the linear case you would have $$\partial_t y = A y$$, where $$A$$ is a matrix. The solution operator $$\mathcal{U}(y) = \exp(A \Delta t)y$$ consists mainly of a matrix exponential. For autonomous systems, this costly matrix exponential evaluation is required only once for the complete propagation -- in contrast to a time-dependent system, where you have to perform this evaluation at every time step.

For nonlinear systems it is not that easy, but depending on the algorithm certain costly evaluations can be re-used.