Self Similar Continuous Solutions. Rarefaction Waves

S.12.4. Self Similar Continuous Solutions. Rarefaction Waves

The transformation ( x, t) → (kx, kt), where k is any positive number, changes neither system (1) nor the initial conditions (6). From the uniqueness of the Riemann problem solution it follows that the unknown u( x, t) depends on a single variable, ξ = x/t. Without loss of generality, we consider the case G(u) = u. The substitution of the self-similar form u( x, t) = u(ξ) into (1) yields

dξ where e F=e F(u) is the matrix with entries F ij = ∂F i ∂u j and I is the identity matrix.

Hence, the velocity vector for the continuous solution u( ξ) is a right eigenvector of the matrix e F

for any point u, and the corresponding eigenvalue equals the self-similar coordinate:

Here, λ k = λ k (u) is a root of the algebraic equation det |e F− λI| = 0, r k =r k (u) is a solution of the corresponding degenerate linear system of equations e F− λI r = 0, and α = α(u) is a positive function, which will be defined below.

Differentiating both sides of the first equation (19) with respect to ξ yields

∂λ k

∂λ k

∂u 1 ∂u n k . n Here, 〈x, y〉 stands for the scalar product of two vectors x and y in the n-dimensional Euclidean

space. Any n× n hyperbolic system allows for n continuous solutions (of system (18)) corresponding to n characteristic velocities λ = λ k . The continuous solutions are determined by n systems of ordinary differential equations. Each system is represented by a phase portrait in the n-dimensional u-space.

A solution/trajectory which corresponds to a characteristic velocity λ k is called a kth rarefaction wave.

Example 8. On calculating the multiplier α for equation (2), one obtains a rarefaction wave expression:

Equations (20) show that the self-similar coordinate ξ is an eigenvalue, which is equal to the tangent to the flux function at the point u = u(ξ); see Fig. 6, where λ = ξ. The trajectory (u(ξ), F (u(ξ)) in the plane (u, F ) lies on the graph of the flux Equations (20) show that the self-similar coordinate ξ is an eigenvalue, which is equal to the tangent to the flux function at the point u = u(ξ); see Fig. 6, where λ = ξ. The trajectory (u(ξ), F (u(ξ)) in the plane (u, F ) lies on the graph of the flux

Figure 8. Mapping from the plane ( x, t) to the hodograph plane (u 1 , u 2 ) and further to the plane of Riemann invariants ( R 1 , R 2 ): (a) characteristics in two centered rarefaction waves; (b) trajectories of two families of rarefaction waves; (c) Riemann invariants are constant along the characteristics and along rarefaction waves for 2 × 2 systems.

function F = F (u). As follows from (20), u = u(ξ) increases in the intervals of concavity of the curve F = F (u), F ′′ ( u) > 0; see Fig. 6.

Let us show that the Riemann invariants are constant along the rarefaction waves for 2 × 2 hyperbolic systems. The substitution of the self-similar solution form u = u( ξ), ξ = x/t, into system (16) results in the following system of two ordinary differential equations:

The equality ξ=λ 1 ( R 1 , R 2 ) takes place along the first rarefaction wave. Hence, the first factor in the second equation of (21) is nonzero. Therefore, the second factor in the second equation of (21) is zero. It follows that R 2 = const along the first rarefaction wave. Along the second rarefaction wave, R 1 is constant. Figure 8a shows two rarefaction wave families that correspond to speeds λ 1 and λ 2 . A continuous solution of a 2 × 2 system u i = u i ( x, t), i = 1, 2, realizes the mapping ( x, t) → (u 1 , u 2 ). The inverse of a characteristic with speed λ i is the curve R i ( u 1 , u 2 ) = const (see Fig. 8b). The expressions R i =R i ( u 1 , u 2 ) realize the mapping ( u 1 , u 2 ) → (R 1 , R 2 ). The inverse of

a characteristic with speed λ i is a set of straight lines that are parallel to the R i -axis (see Fig. 8c).

Example 9. For an adiabatic gas flow [see system (3)–(4)], the rarefaction waves are found from (19) by calculating the right eigenvectors of the matrix A(u) and the function α(u) (see Example 4) to obtain

dρ

dv

( + 22 p = p(ρ). )

Here, the upper sign corresponds to the first eigenvalue and the lower sign, to the second eigenvalue. Eliminating ξ from system (22), we obtain the first-order separable equation

Integrating (23) yields

Z√

p ′ ( ρ) dρ = const .

The left-hand side of (24) taken with the minus sign is equal to the second Riemann invariant, and that taken with the plus sign is equal to the first Riemann invariant. Hence, the second Riemann invariant is constant along the first rarefaction wave and the first Riemann invariant is constant along the second rarefaction wave.

The expressions for Riemann invariants for an ideal polytropic gas are given by formula (17). The trajectories of the rarefaction waves are given by the lines where the Riemann invariants are constant. Figure 7 presents the rarefaction waves for the first characteristic speed (solid lines), where the second Riemann invariant is constant. The dashed lines show the rarefaction waves of the second characteristic speed, where the first Riemann invariant is constant. The arrows show the directions of increasing the self-similar coordinate. Both p v and ρ increase along the first rarefaction in the direction shown in

also increases. Along the second rarefaction, v increases and ρ decreases, and, hence, the second eigenvalue λ = v−

Fig. 7, and the first eigenvalue λ 1 = v+

Aγρ γ−1

Aγρ γ−1 increases.

References for Subsection S.12.4: P. Lax (1973), G. B. Whitham (1974), A. Jeffrey (1976), F. John (1982), B. L. Rozhde- stvenskii and N. N. Yanenko (1983), R. Courant and R. Friedrichs (1985), R. J. LeVeque (2002).

u - Fu ()

arctan D

Figure 9. Illustration to Rankine–Hugoniot and Lax conditions for a shock wave in a scalar conservation law (2).