Evolutionary Shocks. Lax Condition (Various Formulations)

S.12.6. Evolutionary Shocks. Lax Condition (Various Formulations)

In general, discontinuities of solutions are surfaces where conditions are imposed that relate the quantities on both sides of the surfaces. For hyperbolic systems in the conservation-law form (1), these relations have the form (25) and involve the discontinuity velocity D.

The evolutionary conditions are necessary conditions for unique solvability of the problem of the discontinuity interaction with small perturbations depending on the x-coordinate normal to the discontinuity surface. For hyperbolic systems, a one-dimensional small perturbation can be represented as a superposition of n waves, each being a traveling wave propagating at a characteristic velocity λ ❧ i . This allows us to classify all these waves into incoming and outgoing ones, depending on the sign of the difference λ ❧ i −

D. Incoming waves are fully determined by the initial conditions,

while outgoing ones must be determined from the linearized boundary conditions at the shock.

We consider below the stability of a shock with respect to a small perturbation. This kind of stability is determined by incoming waves. For this reason, we focus below on incoming waves.

respectively. It can be shown that if the relation

(31) holds, the problem of the discontinuity interaction with small perturbations is uniquely solvable.

m − + m −1= n

Relation (31) is called the Lax condition. If (31) holds, the corresponding discontinuity is called evolutionary; otherwise, it is called nonevolutionary. For evolutionary discontinuities, small in- coming perturbations generate small outgoing perturbations and small changes in the discontinuity velocity.

For a single equation (2), it follows from (31) that the two waves on both sides of an evolutionary discontinuity must be incoming.

If

+ + m − m −1> n,

then either such discontinuities do not exist or the perturbed quantities cannot be uniquely determined (the given conditions are underdetermined).

If

+ + m − m −1< n,

then the problem of the discontinuity interaction with small perturbations has no solution in the linear approximation. Previous studies of various physical problems have shown that the interaction of nonevolutionary discontinuities with small perturbations results in their disintegration into two or more evolutionary discontinuities.

The evolutionary condition (31) can be rewritten in the form of inequalities relating the shock speed

D and the velocities λ ♠ i of small disturbances. Let us enumerate the characteristic velocities on both sides of the discontinuity so that

λ 1 (u) ≤ λ 2 (u) ≤ ···≤λ n (u).

A shock is called a k-shock if both kth characteristics are incoming:

if

i > k, then D < λ i ♠ ;

if

i < k, then D > λ ♠ i ;

if

i = k, then λ + i < D<λ − i .

Below is another, equivalent statement of the Lax condition: n + 1 inequalities out of the 2n inequalities

(33) must hold.

D≤λ k

( k = 1, . . . , n)

Example 12. The Lax condition (33) for a single equation (2) takes the form

u − )≤ D≤F ′ ( u ).

From the Rankine–Hugoniot condition for one scalar equation (26) it follows that the shock speed D on the plane (u, F ) is equal to the slope of the line segment connecting the “plus” and “minus” points. The graphical interpretation of condition (34) in the plane ( points with coordinates ( u − ,

u, F ) is as follows: the slope of the segment connecting the

F (u) at the point (u , − F (u )) and greater than the slope of

− F (u )) and ( u + ,

)) is less than the slope of the flux curve F (u

F (u) at the point (u ,

)) (see F (u Fig. 9).

Example 13. The Lax condition for the adiabatic gas flow equations (3)–(4) are obtained by substituting the eigenvalue expressions λ=v ♥ √ p ′ ( ρ) (see Example 4) into inequalities (33). We have

The shock evolutionarity requires that three of the four inequalities in (35) hold. Substituting the equation of state for a polytropic ideal gas, p = Ar γ , into (35), we obtain the following evolutionarity criterion:

+ ) γ−1 < v − Aγ(ρ D<v ♥ p

Aγ(ρ − ) γ−1 .

D of the first family decreases from ), for points u + tending to point u − , to v − − + as ρ +

λ 1 (u −

→ 0 and v + → −∞. Along the locus of the second-family shocks, the

∞ as ρ + → ∞ and v → −∞. Figure 10b depicts the locus of points u that can be connected by a shock to the point u + . The evolutionary part of

speed decreases from λ 2 (u ), for points u tending to u −

, to −

the locus is shown by a solid line; the dashed line shows the nonevolutionary part. The shock speed increases from λ 1 (u + ), for points u − tending to u + , to

D of the first family

+ − ∞ as ρ → ∞ and v shocks, the speed increases from → ∞. Along the locus of the second family λ 2 (u ), for points u tending to u + , to v + as ρ − −

→ 0 and v → ∞. References for Subsection S.12.6: O. A. Oleinik (1957), I. M. Gelfand (1959), P. Lax (1973), A. G. Kulikovskii (1979),

C. M. Dafermos (1983), B. L. Rozhdestvenskii and N. N. Yanenko (1983), L. D. Landau and E. M. Lifshitz (1987), D. J. Logan (1994), E. Godlewski and P.-A. Raviart (1996), A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov (2001).