Recipe

Evaluate a rarefaction fan at a point

Use when: you have a rarefaction connecting u_L to u_R (Lax-inadmissible ordering f'(u_L)<f'(u_R)) and need u(x,t) at a specific point

Goal: the correct piece of the fan's piecewise formula, u_L, u_R, or the self-similar profile, for this exact (x,t)

Taught in Rarefaction waves

1. 1

   Compute f'(u_L) and f'(u_R); these mark the two edges of the fan's own wedge in (x,t)-space.

   Example

   For Burgers with u_L=-1,u_R=2: f'(u)=u, so the wedge boundaries are f'(u_L)=-1 and f'(u_R)=2.

   Why: the piecewise formula's three cases are defined entirely in terms of these two boundary slopes.

2. 2

   Divide the target point's x by its t; this single ratio is what the piecewise formula actually branches on.

   Example

   At (x,t)=(1,2): x/t=0.5.

   Why: the fan is self-similar, meaning u depends on x and t only through this one combined variable.

3. 3

   Locate x/t against the two boundaries from Step 1 to decide which of the three cases applies.

   Example

   x/t=0.5 falls in (-1,2), so the middle case applies here.

   • x/t\le f'(u_L) — u=u_L, the constant left state.
   • f'(u_L)<x/t<f'(u_R) — use the self-similar formula u=(f')^{-1}(x/t).
   • x/t\ge f'(u_R) — u=u_R, the constant right state.

   Why: only the middle case actually uses the fan's own formula; the outer two are just the original states, unchanged.

4. 4

   For the middle case, invert f' at x/t; for an outer case, simply report the corresponding constant state.

   Example

   For Burgers, (f')^{-1} is the identity, so u(1,2)=x/t=0.5, directly between u_L=-1 and u_R=2.

   Why: (f')^{-1} is only defined, and only needed, inside the wedge; outside it the solution never varies with x/t.