Riemann's Problem & Applications
Summary
We continue our discussion with Riemann’s problem, the initial value problem $$ \left\{ \begin{alignedat}{2} u_t + (F(u))_x &= 0 \quad&&\text{in}~\mathbb{R}\times(0,\infty)\\ u&=g \quad &&\text{on}~\mathbb{R}\times\{t=0\} \end{alignedat} \right. $$ where $g$ is a piecewise-constant function $$ g(x)=\begin{cases} u_L,& x<0 \\ u_R,& x>0 \end{cases} $$ with $u_L\neq u_R$. Assuming that $F$ is uniformly convex and $C^2$, we give the unique entropy solution of Riemann’s problem for the cases $u_L < u_R$ and $u_L > u_R$.
We then give several examples with more general, piecewise-constant initial data.