My arbitrary order FEM code (found here) was used to solve the Wave Equation with dissipation:

$$ \frac{\partial^2 u}{\partial t^2} + a \frac{\partial u}{\partial t} = c \nabla^2 u \,, $$

where is the displacement of the wave. controls the dissipation of the wave over time and controls the wave propogation speed. The second derivative in time was handled with a finite difference scheme:

$$ \frac{\partial^2 u}{\partial t^2} \approx \frac{u_{i+1} – 2u_i + u_{i-1}}{\Delta t^2} \,. $$

Standard time integration procedures can now be applied.

In the video below, the second order Crank-Nicholson method and a fourth order FEM discretizations were used for space and time, respectively. The video shows the relaxation of the initial sinusoidal perturbation over time in an L shaped domain.