The wave equation is used over a one-dimensional domain
(between and ). The domain is divided into 128 linear elements.
At time , over the entire domain.
After time , at the scalar is prescribed to hold the value and
at the rate of the scalar s () is prescribed to have the
value
.
This disturbance at the right edge propagates into the domain with
the speed of sound ().
At time the rate of over the entire domain is monitored;
at this time point should have become
in
the right half of the domain, whereas nothing should have happened yet in the
left half of the domain.
The first plot shows 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.