In this project, consider the wave equation as a system of first$-$order equations, for an introduction and the DG discretization,

$\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{t}}+$gradp$=0$

$\frac{1}{c^2}\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{t}}+grad*v=0.$

This system is closed by initial conditions for $v(x,0)$ and $p(x,0)$ and one of the following two types of boundary conditions:

$p(x,t)=p_D(x,t)$ for xindel${}_D(Dirichlet$ boundary conditions$),$

$v(x,t)*hat(n)-\frac{1}{c}p(x,t)=0$ for xindel${}_A(absorbing$ boundary conditions$).$

The absorbing boundary condition models "open" boundaries on bounded computational domains.

Tasks

Find an approximate formula for the maximum time step in terms of the prescribed Courant number $($describing the CFL condition$),$ the mesh size $h,$ the polynomial degree $k,$ the density $,$ and the speed of sound $c$ that you can use in your program.

Hint: The maximal transport speed is the largest eigenvalue of the flux matrix