# PDEs appropriate for adaptive time stepping algorithms

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.

• 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. – David Ketcheson Nov 1 '15 at 4:41

Predator-Prey could have the properties you want. Wikipedia has a good description, https://en.m.wikipedia.org/wiki/Lotka–Volterra_equations

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

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.

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