Riemann problems and rarefaction waves

Take $u_0=u_L$ for x<0 and $u_0=u_R$ for x>0, and suppose $f\in C^2$ with $f''>0$ on a neighborhood of the state interval. The initial pattern and equation are unchanged by simultaneous positive scaling of x and t, so the entropy solution is self-similar for t positive. Substituting $u=V(x/t)$ into the smooth equation yields the factor $f'(V)-x/t$ multiplying the derivative of V. A nonconstant smooth fan must therefore satisfy the inverse characteristic-speed relation.

Strict convexity makes the flux derivative strictly increasing and invertible on the relevant interval. When the left state is smaller, characteristic speeds spread from the left endpoint speed to the right endpoint speed. The fan fills that gap continuously and matches the constant states at its edges. This is not a linear interpolation in space at a fixed arbitrary width: its width grows linearly in time, and for a general convex flux its state profile need not be linear in the similarity coordinate.

When the left state exceeds the right state, the characteristics would overlap. The entropy Riemann solution instead has a single shock moving at the flux secant speed. Strict convexity places that speed between the two endpoint characteristic speeds, giving compression. Equal left and right states give a constant solution. These cases cover the strictly convex scalar Riemann problem, but a nonconvex flux can produce compound shock-fan patterns that require a more detailed construction.

The value assigned exactly on a shock curve does not change the weak solution as a space-time equivalence class, since that curve has measure zero. Its traces and speed are what enter conservation and entropy. At a fan edge, by contrast, the displayed rarefaction joins continuously to its constant neighbor. Keeping this distinction avoids interpreting a chosen plotting convention at a discontinuity as an additional physical state or an extra condition in the weak formulation.

At a fixed interface at the origin, evaluate the self-similar solution at similarity coordinate zero. A positive-speed shock leaves the left state at that interface; a negative-speed shock leaves the right state. A rarefaction whose edge speeds straddle zero supplies a sonic interior state with zero characteristic speed. In Burgers' equation that state is zero. Taking an arithmetic average of the endpoint states can therefore select the wrong interface flux even though the formula appears symmetric.

This exact local solution motivates Godunov's numerical flux. For a convex scalar flux, an increasing jump uses the minimum flux on the state interval, while a decreasing jump uses its maximum. The rule is tied to the entropy Riemann solution and the orientation of wave speeds; it is not a universal componentwise formula for systems. In a finite-volume method further questions of step size, reconstruction, and time integration remain before a complete discrete entropy or convergence result can be claimed.