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- #1

Consider the quasi-linear 1-D wave equation

$$

\frac{\partial\rho}{\partial t} + 2\rho\frac{\partial\rho}{\partial x} = 0

$$

with the piecewise constant initial conditions

When $x_0 = \pm 1$

$$

\rho(x,0) = \begin{cases}

4, & x < -x_0\\

3, & -x_0 < x < x_0\\

1, & x > x_0\\

\end{cases}

$$