Riemann problem of non-homogeneous Burgers equation $u_t+uu_x=u$

Question

How to solve $u_t+uu_x=u$ with initial condition $u(x,0)=u_l$ if $x<0$ and $u(x,0)=u_r$ if $x>0$ with $u_l$ and $u_r$ being constant?

Answer 1

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$

$\dfrac{du}{ds}=u$ , letting $u(0)=u_0$ , we have $u=u_0e^s=u_0e^t$

$\dfrac{dx}{ds}=u=u_0e^s$ , letting $x(0)=f(u_0)$ , we have $x=f(u_0)+u_0(e^s-1)=f(ue^{-t})+u(1-e^{-t})$ , i.e. $u=e^tF(x+u(e^{-t}-1))$

$u(x,0)=\begin{cases}ul&\text{if}~x<0\\ur&\text{if}~x>0\end{cases}$ :

$F(x)=\begin{cases}ul&\text{if}~x<0\\ur&\text{if}~x>0\end{cases}$

$\therefore u(x,t)=\begin{cases}ule^t&\text{if}~x<0\\ure^t&\text{if}~x>0\end{cases}$