Solutions for the Riemann Problem

S.12.7. Solutions for the Riemann Problem

In this section, we consider system (1) having a special form, with G(u) = u. The solution of the corresponding Riemann problem (1), (6) is self-similar:

(37) The substitution of (37) into system (1) leads to the system of ordinary differential equations

u = u( ξ),

ξ = x/t.

(18) with the following boundary conditions:

u →u L as ξ → −∞,

u →u R as ξ → ∞.

A trajectory of solution (37) in the space u = ( u 1 , ...,u n ) T is called a solution path. The path is parametrized by the self-similar coordinate ξ. The path connects the point u = u L with the point u=u R . The self-similar coordinate ξ monotonically increases along the path varying from −∞ at u=u L to + ∞ at u = u R . The path consists of continuous segments representing solutions of the ordinary differential equation (18) (rarefaction waves), line segments that connect two points u − and u + satisfying the Rankine–Hugoniot conditions (25) and evolutionary conditions (33), and rest points u( ξ) = const.

Consider an example of a solution consisting of two shocks and one rarefaction. The structural formula* for the solution path is u L →1—2→u R ; specifically,

The shock speed D 1 (resp., D 2 ) can be found from the Hugoniot condition by setting u − =u L and u + =u 1 (resp., u − =u 2 and u + =u R ). Points 1 and 2 are located on the same rarefaction curve. The vector u (2) ( ξ) is a second-family rarefaction wave, which is described by the system of ordinary

differential equations (18) with ξ=λ 2 (u).

Figure 11a depicts a sequence of rarefactions and shocks in the plane ( x, t). Figure 11b shows the profile of the solution component u i along the x-axis. The self-similar curves u = u(ξ) coincide with the profiles u( x, t = 1). For t > 1, the graphs of u(x, t) are obtained from the self-similar curves by extending them along the axis x by a factor of t.

Example 14. Let us discuss the solution to the Riemann problem for a single equation (2) for various forms of the flux function. For concave flux function, with F ′′ ( u) > 0, any shock u L →u R with u L > u R satisfies the Lax condition (34). Hence, the solution to the Riemann problem (2), (6) is given by

u(x, t) =

* In structural formulas like u L →1—2→u R , the symbol “ →” stands for a shock wave and “—” stands for a rarefaction.

0 x () b u i

Figure 11. Solution for the Riemann problem: (a) centered waves in the ( x, t) plane; (b) the u i profile.

Figure 12. Graphical solution to the Riemann problem for a single conservation law (2).

For convex flux function, F ′′ ( u) < 0, any shock u L →u R with u L > u R does not satisfy the Lax condition (34). The solution to the Riemann problem is given by a rarefaction wave:

if − ∞ < x/t < D L = F ′ ( u L ),

u(x, t) =

x/t = F ′ ( u) if D L < x/t < D R = F ′ ( u R ),

if D R < x/t < ∞.

Note that here the solution in the intermediate region is defined implicitly: x/t = F ′ ( u). For convex flux function, F ′′ ( u) < 0, the solution to the Riemann problem with u L > u R is given by a rarefaction wave; the solution for the case u L < u R is given by a shock u L →u R . The solution to the Riemann problem (2), (6) for arbitrary flux function corresponds to the convex envelope of the curve

F (u) inside the interval [u L , u R ] for the case u L < u R . Shocks correspond to line segments between tangent points (e.g., points 1 and 2, 3 and 4, u ′ L and 5, 6 and 7, and 8 and u ′ R in Fig. 12). Rarefactions correspond to segments of the density function between tangent points (e.g., points u L and 1, 2 and 3, and

4 and u R in Fig. 12). The solution to the Riemann problem can be expressed by structural formulas where an arrow stands for a jump and a dash stands for a rarefaction wave. The solution for the case u L < u R , which corresponds to a convex envelope, in Fig. 12 can be expressed by the following structural formula: u L —1 →2—3→4—u R . The solution is given by

∞ < x/t < F

′ ( u L ),

   g(x/t), F ′ ( u L )< x/t < F ′ ( u 1 ),

u(x, t) =

g(x/t), F ′ ( u 1 )= F ′ ( u 2 )< x/t < F ′ ( u 3 ),

 ′   g(x/t), F ( u 3 )= F ′ ( u 4 )< x/t < F ′ ( u R ),

F ′ ( u R )< x/t < ∞,

where the function u = g(ξ) is determined by the inversion of the relation ξ = F ′ ( u). For the case u ′ L > u ′ R , the solution corresponds to the concave envelope (Fig. 12). The corresponding structural formula is: u ′ L →5—6→7—8→u ′ R .

Example 15. The solution to the Riemann problem for strictly hyperbolic systems of two equations with arbitrary initial data can be obtained graphically from the phase portrait for two families of rarefactions (Fig. 7) and for loci of shocks (Figs. 10a and 10b). There are four types of solutions shown in Fig. 13 and outlined below.

Figure 13. Four different cases for the evolution of a discontinuity in gas dynamics; point + L has the coordinates (v − , ρ − + ) and points R n have the coordinates ( v , ρ ).

impermeable membrane

Figure 14. Shock tube problem: the initial distributions of the gas velocities and densities in the tube.

() a t

() b

Figure 15. Decay of density discontinuity in the shock tube: (a) the gas density on the left is lower than that on the right; (b) the gas density on the right is lower than that on the left.

1 ◦ . If the right point R, with coordinates (v + , ρ + ), lies above the locus of the second rarefaction and below the first rarefaction ( R=R 1 ), the solution is given by two rarefaction waves: L—M 1 — R 1 , where M 1 is the intersection point of the loci of

the rarefactions through points L and R 1 .

2 ◦ . If point R lies above both the locus of the first rarefaction and that of the second shock (R = R 2 ), the solution is given by the second shock and the first rarefaction: L→M 2 — R 2 , where M 2 is the intersection point of the locus of the first rarefaction that passes through point R 2 and of the locus of the second shock through point L.

3 ◦ . If point R is located below both the locus of the second rarefaction and that of the first shock (R = R 3 ), the solution is given by the second rarefaction and the first shock: L—M 3 →R 3 , where M 3 is the intersection point of the locus of the second rarefaction that passes through point L and the locus of the first shock through point R 3 .

4 ◦ . If point R lies below the locus of the second shock and above the locus of the first shock (R = R 4 ), the solution is given by two shocks: L→M 4 →R 4 , where M 4 is the intersection point of the shocks loci that pass through points L and R 4 . This solution is given by

(ρ L

if − ∞<ξ<D 1 ,

(v L

if − ∞<ξ<D 1 ,

v R 4 if D 2 < ξ < +∞, where ξ = x/t; the shock speeds D 1 , D 2 and the intermediate point ( ρ M 4 , v M 4 ) are calculated from the Hugoniot conditions

in the form (28) or (29). Problem 1. Let us consider the so-called shock tube problem (see Fig. 14). An impermeable membrane separates the

two parts of the tube and it is suddenly removed at the time t = 0. The gas is at rest in the initial state, v L = v R = 0. The sequence of a shock and a rarefaction on the plane ( x, t) is shown in Fig. 15a for the case ρ L < ρ R . It corresponds to the case where the pressure in the tube on the left ( x < 0) is lower than that on the right (x > 0). The shock races into a quiescent low two parts of the tube and it is suddenly removed at the time t = 0. The gas is at rest in the initial state, v L = v R = 0. The sequence of a shock and a rarefaction on the plane ( x, t) is shown in Fig. 15a for the case ρ L < ρ R . It corresponds to the case where the pressure in the tube on the left ( x < 0) is lower than that on the right (x > 0). The shock races into a quiescent low

Figure 16. Constant velocity piston motion in a tube.

Figure 17. Solution of the piston problem: (a) shock wave in the ( x, t) plane; (b) shock locus in the phase plane. pressure gas. The solution is of the type 2 ◦ above:

  ρ  L if 0 < ξ < D,

if 0 < ξ < D,

   1   v 1 if D<ξ<v 1 + Aγρ γ−1 1 ,

if D<ξ<v 1 +

Aγρ γ−1 1 ,

 eρ(ξ) if v 1 +

 ev(ξ) if v 1 +

Aγρ  γ−1 1 < ξ<v R + Aγρ γ−1

ρ R if v R +

Aγρ γ−1 R < ξ < +∞, where ( ρ 1 , v 1 ) is the intersection point of the locus of the first rarefaction passing through point R and that of the second

shock passing through point L, and the functions eρ = eρ(ξ) and ev = ev(ξ) are determined by solving the algebraic equations

ξ = ev +

Aγeρ γ−1 ,

ev −

Aγeρ γ−1 = v R −

The functions eρ(ξ) and ev(ξ) can be represented in explicit form. A type 3 ◦ solution occurs in the case v L = v R = 0 and ρ L > ρ R . The wave motion is shown in Fig. 15b and the solution is given by 

ρ= eρ(ξ) if v L −

Aγρ γ−1 < ξ<v 1 − Aγρ L γ−1 ev(ξ) if v L  

D < ξ < ∞, where ( ρ 1 , v 1 ) is the intersection point of the locus of the second rarefaction passing through point L and that of the first

shock through point R, and the functions eρ = eρ(ξ) and ev = ev(ξ) are determined by solving the algebraic equations

. Problem 2. Now let us discuss an adiabatic gas flow in a tube in front of an impermeable piston moving with a velocity v L

ξ = ev −

Aγeρ γ−1 ,

ev + 2

Aγeρ γ−1 = v L +

(see Fig. 16). The initial state is defined by prescribing initial values of the velocity and density:

The piston is impermeable; therefore, the gas velocity in front of the shock is equal to the piston velocity (Fig. 17a):

v=v L at ξ=v L .

The gas density in front of the piston is unknown in this problem. Figure 17b shows the locus of points that can be connected by a shock to the point ( v R = 0, ρ R ). This locus is a first-family shock. The intersection of the locus with the line v=v L defines the value ρ L . Hence, ρ L can be found from the equation

= p(ρ L

p(ρ R ρ L ρ R

D=

The shock speed exceeds the piston velocity of (41) for ρ L > ρ R . Both characteristics of the first family as well as the characteristic ahead of the shock from the second family arrive at the shock, so that the Lax condition is satisfied. It can be proved that there are no other configurations that satisfy the initial-boundary value conditions (38), (39). q✂r

References for Subsection S.12.7: I. M. Gelfand (1959), P. Lax (1973), T. P. Liu (1974), C. M. Dafermos (1983, 2000), B. L. Rozhdestvenskii and N. N. Yanenko (1983), J. Smoller (1983), H. Rhee, R. Aris, and N. R. Amundson (1986, 1989), D. J. Logan (1994).