Finding the exact solution of the following Riemann problem. $$\begin{cases} u_t+(e^u)_x=0 ,\\ u(x, 0)=\begin{cases} 1, & x<0, \\ 0, & x>0. \end{cases} \end{cases}$$ I have learned how to solve the advection equation, such as $$ \vec{u}_t+A\vec{u}_x=0, $$with $\vec{u}(x,t)\in\mathbb{R}^m$ and $A\in \mathbb{R}^{m\times m}$.(Solving Linear Hyperbolic System)
But I'm confusing about how to solving this Riemann problem. Any hints or bibliography will be appreciated.
The flux $F:u\mapsto e^u$ of the present conservation law $u_t + F(u)_x =0$ is convex. Therefore, the methodology described in this post applies. Rewrite the conservation law in quasi-linear form $u_t + A(u)u_x =0$ to introduce the characteristic speed $A(u)=e^u$. Here, the base characteristic lines intersect ($A(1) > A(0)$): a shock-wave solution is obtained. Its speed $s$ is given by the Rankine-Hugoniot condition $$ s = \frac{F(1) - F(0)}{1-0} = e-1\, . $$ The entropy solution reads $$ u(x,t) = \left\lbrace \begin{aligned} &1 &&\text{if}\quad x<st ,\\ &0 &&\text{if}\quad x>st . \end{aligned} \right. $$ Note that for the linear advection equation $u_t + Au_x = 0$, the solution would also be a discontinuity, which speed equals the characteristic speed $s=A$.
