I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.

Suppose that the coefficient $c(x)$ in the 1D wave equation $u_{tt}-c(x)^2u_{xx}=0$ has constant values $c_0$ and $c_1$ to the left and right respectively of a bounded interval $I=(x_0,x_1)$, but is variable and nonsmooth on $I$. More precisely, suppose that $c$ is piecewise $C^1$ on $I$ (so that it may have a finite number of jump discontinuities, for example). Consider initial conditions corresponding to a left or right-moving unit impulse at a source point $x_s$: $u(x,0)=\delta(x-x_s)$, $u_t(x,0)=\pm\delta^\prime(x-x_s)$. One wants to compute the (distributional) solution $u(x_r,t)$ for $0<t<T$ at some other point $x_r$. (It may be assumed that $x_s$ is a point of continuity of $c$.)

1) What established numerical techniques are applicable to this problem?

2) What would qualify as the "gold standard" against which new methods should be judged?