Example 11: Propagation of a disturbance in the wave equation.

The wave equation is used over a one-dimensional domain (between $x=0$ and $x=1$). The domain is divided into 128 linear elements. At time $t=0$, $s=0$ over the entire domain. After time $t=0$, at $x=0$ the scalar $s$ is prescribed to hold the value $0$ and at $x=1$ the rate of the scalar s ($\dots$) is prescribed to have the value $1. \; 10 ^{-4}$. This disturbance at the right edge propagates into the domain with the speed of sound ($c=1$). At time $0.5$ the rate of $s$ over the entire domain is monitored; at this time point $\dots$ should have become $1. \; 10 ^{-4}$ in the right half of the domain, whereas nothing should have happened yet in the left half of the domain. The first plot shows $\dots$ if we use purely explicit time stepping (control_timestep_iterations is set to 1).

It is clear that the disturbance did propagate into half of the domain, but quite some oscillations do show up. The oscillations are greatly reduced if two iterations are used (control_timestep_iterations is set to 2); see the second plot.