This problem concerns the shallow water equations
$\partial_tu+u\partial_xu+g\partial_xh=0$
$\partial_th+u\partial_xh+h\partial_xu=0$.
where $g$ is a constant. Water of depth $h_0$ is at rest for $x > 0$, held back by a vertical dam at
$x = 0$. At $t = 0$, the dam breaks, and water pours into the region $x<0$
I have shown that the Riemann invariants $F_{\pm}=u\pm2c$, where $c=\sqrt{gh}$, are constant along characteristics with $\frac{dx_{\pm}}{dt}=u\pm c$. I have also shown that both the $+$ and $-$ characteristics are straight lines for $x>\sqrt{gh_0}t$.
I am struggling with this part
Consider a point $(x_p,t_p)$ immediately to the left of the line $x=\sqrt{gh_0}t$, so that the $x_-$ characteristic began in the region $x>\sqrt{gh_0}t$.
I need to show that the $x_+$ characteristic passing through this point is a straight line passing through the origin, and from this fact I need to show that $u(x_p,t_p)=\frac{2}{3}\left(\frac{x_p}{t_p}-\sqrt{gh_0}\right)$ and $c(x_p,t_p)=\frac{1}{3}\left(\frac{x_p}{t_p}+2\sqrt{gh_0}\right)$.
It makes sense for the $x_+$ characteristic to pass through the origin since there is no water at $x<0$ at the start, whereas if the characteristic started at $x>0$ then there would be crossing of $x_+$ characteristics. This forces $x_+(0)=0$. I don't know how to show that it is a straight line.
As for the part where I show $u(x_p,t_p)$ and $c(x_p,t_p)$ I have tried many different approaches for the past three hours and can't get anywhere.
Best Answer
The dam-break problem is a particular Riemann problem for the shallow water equations in which the initial velocity $u(x,0)$ equals zero. Here, we consider the initial jump in water height $h$ with the values $h(x,0) = 0$ for negative abscissas and $h(x,0) = h_0>0$ for positive abscissas. A picture of the "minus" and "plus" base characteristics is shown below:
With the notation $c=\sqrt{gh}$, the first family transports constant values of $R_- = u-2c$ at the speed $\lambda_- = u-c$, and the second family transports constant values of $R_+ = u+2c$ at the speed $\lambda_+ = u+c$. In the regions $x<0$ and $x>c_0 t$, the method of characteristics provides the values of $$u = \tfrac12(R_+ + R_-), \qquad c = \tfrac14(R_+ - R_-) =\sqrt{gh}$$ which are deduced from the initial values of the Riemann invariants $R_\pm = 0$ and $R_\pm = \pm 2c_0$, respectively. It remains to determine what happens for $0< x < c_0 t$. From R.J. LeVeque (Cambridge University Press, 2002 -- Chap. 13)
Obviously, we should consider a plus-rarefaction wave here (the plus-curves separate at $x=0$). If we place ourselves in the rarefaction fan at $(x_p, t_p)$ far away from the minus-shock (i.e., close to the line $x = c_0 t$), we can state that there is a plus-integral curve passing there. The later follows from the Ansatz $R_\pm = \tilde R_\pm(\xi)$ with $\xi = x/t$, and the relationship $$ \left(\lambda_\pm - \xi\right) \tilde R'_\pm(\xi) = 0 $$ is deduced from the transport equations $\partial_t R_\pm + \lambda_\pm \partial_x R_\pm = 0$. For a plus-rarefaction wave along which $R_+$ varies, we therefore know that $\xi = \lambda_+$, and that $R_-$ is constant. Since $R_-$ is constant and equal to its value $-2c_0$ at some nearby point located on the line $x = c_0 t$, we can write the system \begin{aligned} u_p + \phantom{1} c_p &= x_p/t_p \\ u_p - 2 c_p &= -2c_0 \end{aligned} which solution may be expressed as $$ u_p = \frac23 \left(\frac{x_p}{t_p} - c_0\right) ,\qquad c_p = \frac13 \left(\frac{x_p}{t_p} + 2c_0\right) . $$