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I'm looking for some physical phenomena for which an adaptive time stepping algorithm would be ideal. A PDE or ODE that showed very large gradients in time at a small period of time and smoother gradients for the rest of the still necessary simulation. Thanks.

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  • $\begingroup$ Adaptive time stepping is useful for almost any nonlinear problem. There are a number of nice ODE and PDE examples where adaptive time stepping is relevant in the two-volume series by Hairer, Norsett, & Wanner. $\endgroup$ – David Ketcheson Nov 1 '15 at 4:41
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Predator-Prey could have the properties you want. Wikipedia has a good description, https://en.m.wikipedia.org/wiki/Lotka–Volterra_equations

Wikipedia predator-prey simulation

Alternatively you could solve very simple decay ODE with an impulse response. You would have three regimes:

  1. Steady state
  2. Impulse, in which the system is perturbed over a small time scale
  3. Decay back to steady state

$$ \frac{dy}{dt} = s(t) - ky $$

Make $s(t)$ sharp and the solution will have a similar shape provided $k$ is large. Here $k=\tau^{-1}$ is the decay rate, the inverse of which can be interpreted as the decay lifetime, how quickly the system returns to equilibrium after the perturbation.

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ODEs modelling chemical reactions (chemical kinetics) are another good test case. When you have multiple reactions with different rates (some reactions are blazing fast, while others take a much longer time) you end up with a stiff system that is a good candidate for adaptive time stepping.

Here is an example from the classic book by Hairer and Wanner: the Brusselator (aka the Brussels oscillator, Chapter II.4, page 170)

\begin{eqnarray} \frac{dy_1}{dx} &=& 1 + y_1^2 y_2 - 4 y_1 \\ \frac{dy_2}{dx} &=& 3y_1 - y_1^2 y_2 \end{eqnarray}

with initial values $y_1(0) = 1.5$, $y_2(0)=3$, $0 \le x \le 20$.

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