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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)?

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  • $\begingroup$ Have you tried discontinuous initial values? $\endgroup$ – daw Feb 22 at 9:20
  • $\begingroup$ I assume you mean weak solutions? $\endgroup$ – Klaus Feb 22 at 9:25
  • $\begingroup$ Yes, I'm looking for weak solutions. I'm not quite sure what you mean with discontinous inital values. If I set (for example) $u= cos(t) * some indicator function$ I would get a discontinous inital condition, but then I have problems to find the right side because of the missing derivatives of this solution $\endgroup$ – Julian H Feb 22 at 13:09

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