Burgers Riemann Entropy Shock and Fan

The inviscid scalar conservation law u_t + (u²/2)_x = 0, known as the inviscid Burgers' equation, provides a classic model for nonlinear wave propagation, shock formation, and rarefaction. A Riemann problem consists of a piecewise constant initial condition with a left state u_L and a right state u_R. Because the flux function f(u) = u²/2 is strictly convex, characteristics will either collide or spread out, leading to two distinct types of solutions.

experiment

We can observe both shock and rarefaction behaviors using the Burgers Riemann Entropy Shock and Fan interactive module.

1. Set the Left state to -1 and Right state to 2. Here u_L < u_R, so characteristics spread apart.
2. Set Time to 1. The characteristics form a rarefaction fan filling the space between x = -t (speed -1) and x = 2t (speed 2).
3. Move the Probe position to 0. Inside the fan, the solution is u = x/t, so the probe reads 0.
4. Change Time to 0. Notice the right-at-zero convention taking effect: the initial step is evaluated, so the probe at x=0 reads 2 instead of an undefined or averaged value.
5. Change both states to 1. This forms a constant state with no shocks or fans. The probe reads 1 everywhere.