Can you construct a discontinous solution for the wave equation $u_{tt} - c^2 \Delta u = f $ with homogenous dirichlet boundary conditions on the domain $\Omega = [0,1]^2$?
Background: I programmed a tool which solves the wave equation in 1d ($u_{tt} - c^2 u_{xx} = f$, $\Omega = [0,1]$) and 2d ($u_{tt} - c^2 ( u_{xx} + u_{yy}) = f$, $\Omega = [0,1]^2$) with dirichlet boundary conditions and a starting deflection $u_0$ and and a starting velocity $v_0$. I use the finite element in space and leapfrog/crank-nicolson/hht-$\alpha$ in time. Therefore I expect a convergence of $\mathcal{O}(h^2 + \Delta t^2)$ if $u\in C^4$. In one dimension, I use d'Alembert to calculate the reference solution. This also works with discontinous solutions. I want to test the convergence speed in 2d if $u \notin C^4 $, for example if $u$ is discontinous. Can I construct a reference solution in 2d to calculate the error ($L_2$-norm)?