= Riemann problem
{c}
{title2=$U(x,t)=U(x/t)$}
A Riemann problem for a one-dimensional <system of conservation laws> $\partial_tU+\partial_xf(U)=0$ has two constant initial states, $U_L$ for $x<0$ and $U_R$ for $x>0$. The data and equation are invariant under $(x,t)\mapsto(\lambda x,\lambda t)$. When the physically admissible <entropy solution> is unique, this invariance makes it a <self-similar solution> depending on $x/t$. For ideal-gas <compressible flow>, the characteristic speeds are $u-c,u,u+c$. The two acoustic families carry <shock waves> or <rarefaction waves>, while the middle family carries a <contact discontinuity>. Sufficiently strong separating velocities can instead create a vacuum region.
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