Order Form - FENOMEC

40 CHAPTER II. THE RIEMANN PROBLEM
This is in strong contrast with Lax shock inequalities which impose that the characteristics
impinge on the discontinuity from both sides. Undercompressive waves are a
potential source of non-uniqueness, as will become clear shortly.
The rarefaction waves associated with the equation (1.1) were already studied in
Section 2. For concave-convex fluxes we found (see (2.8)):
⎧ [
⎪⎨ u− , +∞ ) , u − > 0,
( )
R(u − )= −∞, +∞ , u− =0,
(3.14)
⎪⎩ ( ]
−∞,u− , u− < 0.
We are now in the position to solve the Riemann problem (1.1), (1.2), and (1.4).
Theorem 3.5. (One-parameter family of Riemann solutions for concave-convex functions.)
Suppose that the flux f is a concave-convex function (see (2.5)) andfixsome
Riemann data u l and u r . Restricting attention to solutions satisfying the entropy
inequality (1.4) for a given strictly convex entropy pair (U, F ) and assuming for definiteness
that u l ≥ 0, the Riemann problem (1.1) and (1.2) admits the following
solutions in the class P:
(a) If u r ≥ u l , the solution is unique and consists of a rarefaction connecting
continuously u l to u r .
(b) If u r ∈ [ ϕ ♯ 0 (u )
l),u l , the solution is unique and consists of a classical shock
connecting u l to u r .
(c) If u r ∈ [ ϕ ♭ 0(u l ),ϕ ♯ 0 (u l) ) , there exist infinitely many solutions, consisting of a
nonclassical shock connecting u l to some intermediate state u m followed by
–aclassicalshockifu m 0 may contain two shocks
and have a total variation which is larger than the one of its initial data.
Proof. The functions described above are clearly solutions of the Riemann problem.
The only issue is to see whether they are the only admissible solutions. The argument
below is based on the two key properties (3.3) and (3.4). We recall that two wave
fans can be combined only when the largest speed of the left-hand wave is less than
or equal to the smaller speed of the right-hand one.
Claim 1 : A nonclassical shock connecting u − to u + ∈ [ ϕ ♭ 0(u − ),ϕ ♮ (u − ) )
can be followed only by a shock connecting to a value u 2 ∈ [ u + ,ρ(u − ,u + ) )
or else by a rarefaction to u 2 ≤ u + .
Indeed, each state u 2 ∈ [ u + ,ρ(u − ,u + ) ) is associated with a classical shock propagating
at the speed a(u 2 ,u + ), which is greater than a(u − ,u + ). These states are
thus attainable by adding a classical shock after the nonclassical one. On the other
hand, a state u 2 ∈ ( ϕ ♯ 0 (u +),ϕ ♭ 0(u + ) ] cannot be reached by adding a second shock
after the non-classical one since, by the property (3.4), ϕ ♭ 0(u + )=u − and therefore