the Riemann problem

Take a conservation law and feed it the simplest interesting initial data imaginable: a single jump. Everything is one constant value to the left of the origin and another constant value to the right, with a clean discontinuity at x = 0. What happens next? This deceptively plain question — the Riemann problem — is the fundamental cell out of which the whole theory of conservation laws is built.

Concretely, solve u_t + f(u)_x = 0 (or a system U_t + f(U)_x = 0) with initial data u_0(x) = u_L for x < 0 and u_R for x > 0. The data has a beautiful symmetry: it is invariant under the rescaling (x, t) -> (a x, a t), so the solution depends only on the single similarity variable xi = x/t. That collapses the PDE to an ODE in xi and makes the answer explicit. For a scalar convex flux the solution is exactly ONE elementary wave: a single shock if the data is compressive (u_L > u_R for Burgers), or a single rarefaction fan if it is expansive (u_L < u_R). For an n-by-n strictly hyperbolic system, the answer is a sequence of n elementary waves — shocks, rarefactions, and contact discontinuities, one per characteristic field — separating n+1 constant states, found by intersecting wave curves through the (u_L, u_R) states.

Why it matters: the Riemann problem is the local atom of hyperbolic theory. Its solution defines admissibility (it is where the entropy condition is tested), it is the model for how any discontinuity in a general solution behaves, and it is literally the engine inside Godunov-type numerical schemes, which advance a simulation by solving a tiny Riemann problem at every cell interface. The classic physical instance is the shock tube: a membrane separating high- and low-pressure gas is burst, and out come a shock, a contact, and a rarefaction — exactly the three-wave structure the Euler Riemann problem predicts.