THE RIEMANN PROBLEM FOR FLUID FLOWS IN A NOZZLE WITH DISCONTINUOUS CROSS-SECTION

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1 THE RIEMANN PROBLEM FOR FLUID FLOWS IN A NOZZLE WITH DISCONTINUOUS CROSS-SECTION Philippe G. LeFloch and Mai Duc Thanh, Abstract. The system of balance laws describing a compressible fluid flow in a nozzle forms a non-strictly hyperbolic systems of partial differential equations which, also, is not fully conservative due to the effect of the geometry. First, we investigate the general properties of the system and determine all possible wave combinations. Second, we construct analytically the solutions of the Riemann problem for any values of the left- and right-hand states. For certain values we obtain up to three solutions whose structure is carefully described here. In some range of Riemann data, no solution exists. When three solutions are available, then exactly one of them contains two stationary waves which are super-imposed in the physical space. We include also numerical plots of these solutions.. Introduction We consider the Riemann problem for the following system describing the evolution of an isothermal fluid in a nozzle with discontinuous cross-sectional area a(x) > : (.) t (aρ) + x (aρu) =, t (aρu) + x (a(ρu + p(ρ))) = p(ρ) x a, t a =, x RI, t >. Here, ρ and u stand for the density and the particle velocity of the fluid under consideration, respectively, and the pressure function p = p(ρ) is given by (.) p(ρ) = κρ γ, < γ < 5/. Observe that the third equation in (.) is trivial since the function a depends only on x. Since we are interested in discontinuous functions a it is convenient (following []) to consider the full set of three equations. The third equation is associated with a linearly degenerate field with constant characteristic speed. First, observe that the system (.) does not take the usual form of a system of conservation laws and is not fully conservative. This is due to the effect of the geometry modeled by the function a. When a admits discontinuities, Dirac masses appear on the right-hand side of the second equation of (.). Therefore, the usual notion of weak solutions for systems of conservation laws does not apply. The product still makes sense as a measure within the framework introduced by Dal Maso, LeFloch, and Murat [5]. Throughout this paper, the function a will be assumed to be piecewise constant, so that the application of the theory in [5] is particularly immediate, as we see below. Centre de Mathématiques Appliquées & Centre National de la Recherche Scientifique, U.M.R. 76, Ecole Polytechnique, 98 Palaiseau Cedex, France. lefloch@cmap.polytechnique.fr. Institute of Mathematics, P.O. Box 6 BoHo, Hanoi, Vietnam. thanh@cmap.polytechnique.fr. Mathematics Subject Classification. Primary: 5L65, 7XX. Secondary: 76N, 76L5. Key words and phrases: compressible fluid, nozzle with discontinuous cross-section, conservation law, non-strict hyperbolicity, shock wave, entropy inequality, Riemann problem. Typeset by AMS-TEX

2 LEFLOCH AND THANH For smooth solutions (x, t) (ρ, u, a), the system (.) is equivalent to the following three conservation laws: t (aρ) + x (aρu) =, (.) t u + x ( u + h(ρ)) =, t a =, where the function h is defined by h (ρ) = p (ρ)/ρ, thus h(ρ) := κγ γ ργ. On the other hand, the Rankine-Hugoniot relation associated with the third equation in (.) takes the form (.) λ[a] =, where λ denotes the speed of the discontinuity, [a] := a + a is the jump of the quantity a, and a ± denotes its left- and right-hand traces. The relation (.) implies that : (i) the component a remains constant across the shock, or (ii) a is discontinuous but the shock velocity vanishes. The following discussion provides us the list of admissible waves for solving the Riemann problem, and is central to this paper. Let us assume first that the component a remains constant across some shock wave. Since a is piecewise constant, it should be constant in a neighborhood of the shock. Eliminating a from (.), we obtain the following system of two conservation laws (.5) t (ρ) + x (ρu) =, t (ρu) + x ((ρu + p(ρ))) =. Thus, the left- and right-hand states are related by the Rankine Hugoniot relations corresponding to (.5) (.6) λ[ρ] + [ρu] =, λ[ρu] + [ρu + p(ρ)] =, where [ρ] := ρ + ρ +, etc. Suppose next that the component a is discontinuous and that, therefore, the shock speed vanishes. The solution is independent of time, ad it is natural to search for a solution as the limit of a sequence of time-independent smooth solutions of (.). Following Marchesin-Paes-Leme [8] and LeFloch [], this motivates us to consider the system of ordinary differential equations (.7) (aρu) =, ( u + h(ρ)) =. The integral curve of (.7) passing through each point (ρ, u, a ) can be parameterized by u, say u (ρ(u), u, a(u)), and satisfie u + κγ γ ργ (u) = u + κγ γ ργ, a(u) = a ρ u ρ(u)u.

3 NOZZLE WITH DISCONTINUOUS CROSS-SECTION Letting u u ± and setting ρ ± = ρ(u ± ), a ± = a(u ± ), we see that the states (ρ ±, u ±, a ± ) satisfy the Rankine-Hugoniot relations associated with (.), but with zero shock speed: (.8) [aρu] =, [ u + h(ρ)] =. It is easy to check (LeFloch []) that discontinuities satisfying (.8) are solutions of the nonconservative system (.) in the sense of DalMaso-LeFloch-Murat for the family of paths based on the ODE trajectories (.7). In this paper, we will construct solutions of the Riemann problem associated with (.), made of elementary waves defined as follows. Definition.. The admissible waves for the system (.) are the following ones: the rarefaction waves, which are smooth solutions of (.) with constant component a depending only on the self-similarity variable x/t; the shock waves which satisfy (.6) and Lax shock inequalities (see []) and have constant component a; and the stationary waves which have zero propagation speed and are given by (.8). As will become clear in Section, the system under consideration in this paper is not strictly hyperbolic. This makes the analysis of the Riemann problem particularly challenging. For previous works in this direction we refer to [,6,9,,,6,7,8]. For numerical work on hyperbolic systems with source-term we refer to [,, 7, 8]. An outline of this paper is as follows: Section presents basic material on the system (.). Then, in Section, for each wave family we construct one-parameter sets of states which can be connected to a given state by using shock and rarefaction waves in one family plus a stationary wave. In Sections and 5, we exhibit the solution (or solutions) of the Riemann problem: first using a single stationary wave (Section ) and, second, using two stationary waves (Section 5). The Riemann solutions are finally plotted numerically in Section 6. Note : The authors gratefully acknowledge the support and hospitality of the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, where this research was performed during the Semester Program Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics. After the completion of this work, the authors received an interesting preprint by N. Andrianov and G. Warnecke On the solution to the Riemann problem for the compressible duct flow, which addresses the existence and uniqueness for the full model of three equations of fluid dynamics in a nozzle. Based on two-dimensional numerical computations, the authors propose an interesting selection criterion which may be relevant to select physically meaningful solutions in the cases where we exhibit multiple solutions in the present paper.. Preliminaries It will be convenient to write U = (ρ, u, a). For smooth solutions the system (.) can be written in the standard form (.) t U + A(U) x U =, where A = h (ρ) u. u ρ ρu/a

4 LEFLOCH AND THANH The matrix A is not the Jacobian of a function. It admits the following three eigenvalues and righteigenvectors: (.) λ := u p (ρ), λ :=, λ := u + p (ρ), r := (ρ, p (ρ), ) t, r := (ρu, p (ρ), a(u p (ρ) u ))t, r := (ρ, p (ρ), ) t. The first and the second characteristic speeds may coincide, and so do the second and the third characteristic speeds. More precisely, setting (.) C ± : u = ± κγ ρ γ, we see that λ = λ on C +, λ = λ on C and (.) is not strictly hyperbolic. In (ρ, u)-plan, the boundaries of strict hyperbolicity C ± separates the half-plane ρ > into three (open) domains. For convenience, we will refer to them as the lower region G, the middle region G, and the upper region G : G = {(ρ, u) : u < p (ρ)}, G = {(ρ, u) : u < p (ρ)}, G = {(ρ, u) : u > p (ρ)}. In each of these regions the system (.) is strictly hyperbolic and we have It will be also convenient to use the notation λ < λ < λ, in G, λ < λ < λ, in G, λ < λ < λ, in G. G + = {(ρ, u) G : u > }, G = {(ρ, u) G : u < }. The second characteristic family is linearly degenerate, while the - and the -characteristic families are genuinely nonlinear: λ r = λ r = p (ρ) (ρp (ρ) + p (ρ)) >. A shock wave in the first or third family connecting a given left-hand state U to a right-hand state U must satisfy the Lax shock inequalities (see []) (.) λ i (U) < λ i (U, U ) < λ i (U ), i =,, where λ i (U, U ) denotes the shock speed. Shock waves from - and -families are constrained by the jump conditions (.6). Hugoniot set is determined (u u ) = κ( ρ ρ )(ργ ρ γ ). The so-called

5 NOZZLE WITH DISCONTINUOUS CROSS-SECTION 5 We can distinguish between two families. Using the condition that the -Hugoniot curve issued from U is tangent to the -eigenvector vector r (U), we find H (U ) : u = ( u κ( ) /, ρ ρ )(ργ ρ γ ) ρ > ρ, ( u + κ( ) /, ρ ρ )(ργ ρ γ ) ρ < ρ. Similar arguments lead to H (U ). Using Lax shock inequalities (.), we can define the -wave and a -wave (left-to-right) shock curve S (U ) and S (U ) consisting of all right-hand states U that can be connected to U by a shock by: (.5) ( S (U ) : ω (ρ; U ) := u κ( ) /, ρ ρ )(ργ ρ γ ) ρ > ρ, S (U ) : ω (ρ) := u ( κ( ρ ρ )(ργ ρ γ ) ) /, ρ < ρ. The condition ρ > ρ for S (U ) (respectively ρ < ρ for S (U )) is derived from the Lax shock inequalities (.). Similarly, the -wave and -wave (right-to-left) shock curve S (U ), S (U ) consisting of all left-hand states U that can be connected to U by a Lax shock are : (.6) ( ω (ρ) := u + κ( ) /, ρ ρ )(ργ ρ γ ) ρ < ρ, ( ω (ρ) := u + κ( ) /, ρ ρ )(ργ ρ γ ) ρ > ρ. Next, we search for rarefaction waves, i.e., smooth self-similar solutions to the system (.). As usual, we consider for ordinary differential equations (.7) du dξ = r i (U), i =,, λ i r i (U) U(ξ ) = U. Therefore, we can determine the -wave (left-to-right) rarefaction curve R (U ) by (.8) ρ p (ρ) ω (ρ) = u dρ ρ ρ = u κγ γ (ρ(γ )/ ρ (γ )/ ), ρ ρ, γ γ (ξ) = ρ (ξ ξ ), ξ ξ. κγ(γ + ) ρ γ and the -wave (right-to-left) rarefaction curve R (U ) by (.9) ρ p (ρ) ω (ρ) = u + dρ, ρ ρ, ρ ρ γ γ (ξ) = ρ (ξ ξ ), ξ ξ. κγ(γ + ) ρ γ +

6 6 LEFLOCH AND THANH If we take ξ ξ and ρ ρ in the formulas (.8), (.9) instead, we get the -wave (right-to-left) and -wave (left-to-right) rarefaction curve R (U ), R (U ). In conclusion, we can define the wave curves (.) W (U ) := S (U ) R (U ), W (U ) := S (U ) R (U ), W (U ) := S (U ) R (U ), W (U ) := S (U ) R (U ). Let us investigate properties of the wave curves. A state U being given we consider the wave curves in the (ρ, u)-plan issuing from U. It follows from (.6) that [ κ ρ (ρ γ ρ γ (.a) = ) + ( <, ρ > ρ, and, clearly, dω (U, ρ) dρ ρ ρ )γργ ] ω (U, ρ) u (.b) Similar calculations show that (.) dω (U, ρ) dρ = κγρ γ <, ρ ρ. dω (U, ρ) dρ >, ρ >. From (.)-(.), we have the well-known property: Lemma.. (- and -wave curves) The wave curve W (U ) : ρ ω (U, ρ), ρ > is strictly decreasing. The wave curve W (U ) : ρ ω (U, ρ), ρ > is strictly increasing. (.) As observed in Section, a stationary shock from a given state U to some state U must satisfy ( W (U ) : u = ω (U, ρ) := sgn(u ) u κγ /, γ (ργ ρ γ )) u, Σ(U, ρ; a) := ω (U, ρ)ρ a u ρ a =. If u =, then the equations (.) determine three points (ρ, ), (, ± (κγ)/(γ )ρ (γ )/ ). Assume u. The function ρ ω (U, ρ) is defined provided the expression under the square root is nonnegative: u κγ γ (ργ ρ γ ), which requires that (.) ρ ρ(u ) := ( γ ) κγ u + ρ γ γ. In the interval [, ρ(u )], the function ρ ω (U, ρ) is decreasing for u > and increasing for u <. On the other hand, (.5) Σ(U, ρ; a) ρ = u κγ γ (ργ ρ γ ) κγρ γ ( ) /. u κγ γ (ργ ρ γ )

7 NOZZLE WITH DISCONTINUOUS CROSS-SECTION 7 Assume, for simplicity, that u >. The last expression means that (.6) where (.7) ρ max (ρ, u ) := Σ(U, ρ; a) >, ρ ρ < ρ max (ρ, u ), Σ(U, ρ; a) <, ρ ρ > ρ max (ρ, u ), ( γ κγ(γ + ) u + ) γ + ργ γ. The function ρ Σ(U, ρ; a) takes negative values at the endpoints. Thus, it admits some root if and only if the maximum value is non-negative. This is equivalent to saying that a ρ u (.8) a a min (U ) := γ+. κγρ max(ρ, u ) For u <, the same properties hold provided we reverse the inequalities (.6) and replace negative by positive. Based on these observations, we have Lemma.. (Waves associated with the linearly degenerate field.) Given U, a stationary shock issuing from U and connecting to some state U = (ρ, u, a) exists if and only if a a min (U ). More precisely, we have : (i) If a < a min (U ), there are no stationary shocks. (ii) If a > a min (U ), then there are exactly two values ϕ (U, a) < ρ max (U ) < ϕ (U, a) such that (.9) Σ(U, ϕ (U, a), a) = Σ(U, ϕ (U, a), a) =. Accordingly, along the curve W (U ), there are two distinct points that can be attained from U using a stationary shock. (iii) If a = a min (U ), then on the curve W (U ) there is a unique point that can be attained from U using a stationary shock. The following lemma compare together terms arising in the previous lemma. Lemma.. i) We have (.) ρ max (ρ, u ) < ρ, (ρ, u ) G, ρ max (ρ, u ) > ρ, (ρ, u ) G G, ρ max (ρ, u ) = ρ, (ρ, u ) C ±. ii) After a stationary jump, the state (ϕ (U, a), ω (U, ϕ (U, a))) belongs to G if u <, and belongs to G if u >, while the state (ϕ (U, a), ω (U, ϕ (U, a))) always belongs to G. Moreover, for u, we have (.) In addition, we have If a > a, then (.a) (.b) If a < a, then (ρ max (U, a), ω (U, ρ max (U, a))) C + if u >, (ρ max (U, a), ω (U, ρ max (U, a))) C if u <. ϕ (U, a) < ρ < ϕ (U, a). ρ < ϕ (U, a) for U G G, ρ > ϕ (U, a) for U G.

8 8 LEFLOCH AND THANH iii) (.) a min (U, a) < a, (ρ, u) G i, i =,,, a min (U, a) = a, (ρ, u) C ±, a min (U, a) =, ρ = or u =. Proof. The inequalities (.) are straightforward. To demonstrate ii) and iii), we can assume for simplicity that u >. Let us define the function Then, a straightforward calculation gives g(u, ρ) = ω (U, ρ) κγρ γ which proves (.). On the other hand, since = u κγ γ (ργ ρ γ ) κγρ γ. dg(u, ρ) dρ g(u, ρ max (U )) =, = (γ + )κγρ γ <, and that ϕ (U, a) < ρ max (U, a) < ϕ (U, a) it holds that g(u, ϕ (U, a)) > g(u, ρ max (U )) = > g(u, ϕ (U, a)). The last two inequalities justify the statement in ii). Moreover, Σ(U, ρ ; a) = ρ u ( a /a) > iff a > a, which proves (.a), and shows that ρ is located outside of the interval [ϕ (U, a), ϕ (U, a)] in the opposite case. It is then derived from (.5) that Σ(U, ρ ; a) ρ = u κγρ γ < iff U G, u which, together with the earlier observation, implies (.b). We next check (.) for a = a. It comes from the definition of a min (U ) that a min (U ) < a if and only if κγρ γ+ > ρ u, that can be equivalently written as where m := ρ γ /u. Then, we can see that l(m) := γ γ m (κγ) γ+ m γ+ + γ + κγ(γ + ) >, (.) l(/κγ) =, which, in particular shows that the second equation in (.) holds, since (ρ, u ) C ± for m = /κγ. Moreover, dl(m) dm = γ ( (κγm) γ+ ), γ + which is positive for m > /κγ and negative for m < κγ. This together with (.) establish the first statement in (.). The third statement in (.) is straightforward. This completes the proof of Lemma..

9 NOZZLE WITH DISCONTINUOUS CROSS-SECTION 9 Proposition.. i) The -shock speed λ (U, U), (for ρ > ρ ) may change sign along the -shock curve S (U ). More precisely, if U G G then λ (U, U) remains negative: (.5a) λ (U, U) <, U S (U ). If U G, then λ (U, U) vanishes once at some point Ũ corresponding to a value ρ = ψ (U ) > ρ on the -shock curve S (U ) : ρ U = (ρ, ω (ρ; U )), such that λ (U, Ũ) =, (.5b) λ (U, U) >, ρ (ρ, ρ ), λ (U, U) <, ρ ( ρ, + ), Ũ G +. ii) The -shock speed λ (U, U), (for ρ > ρ ) may change sign along the -shock backward curve S (U ). More precisely, if U G G then λ (U, U) remains positive: (.6a) λ (U, U) <, U S (U ). If U G, then λ (U, U) vanishes once at some point Ũ corresponding to a value ρ = ψ (U ) > ρ on the -shock backward curve S (U ) : ρ U = (ρ, ω (ρ; U )), such that λ (U, Ũ) =, (.6b) λ (U, U) <, ρ (ρ, ρ ), λ (U, U) >, ρ ( ρ, + ), Ũ G. Proof. We only prove i), since similar arguments can be used for ii). From (.6) and (.6) we deduced that (.7) λ (U, U) = ρω (U, ρ) ρ u ( = u κ ρ ρ γ ρ γ ) /. ρ ρ ρ ρ ρ Thus, if U G C, then u < and λ (U, U) <. If U G, then, since the function ρ ρ γ is convex and ρ > ρ, we deduce from (.) that λ (U, U) < u κγρ γ <, where the last inequality holds by U G. Letting U C + in the last inequality, we obtain Thus, (.5a) is established. Assume now that U G. Set λ (U, U) <, U U, U C +. (.8) l(ρ) := κ ρ ρ ρ γ ρ γ ρ ρ u =, ρ ρ. Observe that the function l(ρ) and λ (U, U) have the same roots and take values of opposite signs. Then, a straightforward calculation shows that l(ρ ) = κγρ γ u <, l(+ ) = +, l (ρ) = κ (γ )ργ + ρ γ ρ (ρ ρ ) >,

10 LEFLOCH AND THANH which implies the existence of the value ψ (U ) as indicated in (.5b). Moreover, the three statements in the beginning of (.5b) are satisfied. In view of the jump relation [ρu] =, for a Lax shock with zero speed, the value ω (U, ψ (U )) must be positive. The fact that the function ρ ρ γ is convex and that the function ω is decreasing in ρ yields = λ ( (U, Ũ) = u κ ρ ρ γ ρ γ ) / ρ ρ ρ > u κγψ (U ) γ > ω (U, ψ (U )) κγψ (U ) γ = λ (ψ (U ), ω (U, ψ (U ))), which, since ω (U, ψ (U )) >, proves the last statement of (.5b). Proposition.. Let us introduce two other curves in the (ρ, u)-plane: This completes the proof of (.9) C : u (ρ) = κγ γ ρ γ, C : u (ρ) = κγ γ ρ γ, Formally, the curve C is given by R (ρ =, u = ), and the curve C is given by R (ρ =, u = ). At this stage, we have: Lemma.5. i) If U is located above the curve C, then the curve W (U ) has a unique intersection with each curve C ±, C and C. If U is located below C, no such an intersection is available. ii) If U is located below or on the curve C, then the curve W (U ) has a unique intersection with each curve C ±, C and C. If U is located above C, no such an intersection is available.. Two-parameter sets of composite waves Solution curves of the Riemann problem for (.) are understood to be either wave curves that were already defined in the previous section or sets of composite waves W, (U, a), W, (U, a) (for - and -wave families) or W, (U, a), W, (U, a) (for - and -wave families). For simplicity, we restrict our attention in this section to the situation that there is only one stationary shock in each composite wave. Two stationary shocks in a composite wave is also possible and this will be discussed later in Section 5. Definition.. The composite curve W i (U, a) is the set of all states U = (ρ, u, a) such that there exists a state U W (U ) and the -wave from U to U can be followed by the stationary shock from U to U by using ϕ i (U, a), i =,. The composite curve W i (U, a) is the set of all the states U = (ρ, u, a) such that the (fixed) stationary jump from U to some U (using ϕ i (U, a)) can be followed by the -waves from U to U, i =,. Under the transformation (.) x x, u u, a right-hand state U = (ρ, u, a) becomes a left-hand state of the form (ρ, u, a). Therefore, without loss of generality, we can always assume from now on that (.) a < a.

11 NOZZLE WITH DISCONTINUOUS CROSS-SECTION Figure.: U below C We need only construct W, (U, a), as similar arguments can be used for other cases. Set (.) U ± := W (U ) C ±, Z(U ) = W (U ) {u = }. First, assume that U is below or on the curve C (Figure.). The curve W (U ) always remains in G and does not cross the strict hyperbolicity boundary C ±. Therefore, all the -waves have negative speeds. Consequently, the -waves can not be followed by -waves. The only way is that the -waves are followed by -waves. Thus, we have in this case (.) W, (U, a) W (U, a) W (U, a). Second, U is between C and C + or on the curve C + (Figure.). The construction can be a -wave from U to U = (ρ, u) W (U ) as long as U do not belong to G, followed by a stationary jump by either using ϕ (U, a) to a state U with (.5) U G if u, U G if u, or using ϕ (U, a) to a state U G. Such states U form W (U, a). The states U G form the set W (U, a). In the case U G reached by a stationary jump from U Ḡ+, we have (.6) λ (U ) >,

12 ( ' ) ) * ) ) LEFLOCH AND THANH /5*,.! * * /,!. $76 * */,!. */,!. $&% "# )+*-,!. Figure.: U between C + and C the construction can therefore be continued with W (U ) as long as the -shock speed from U is nonnegative. This is W (U, a) with any U belongs to W (U ) between U + and Z(U ). Hence, we have in this case two curves and a one-parameter family of solutions described by (.7) W, (U, a) W (U, a) W (U, a) W (U, a) U W (U ), U between U + and Z(U ). Finally, assume that U is above C +, i.e, U G (Figure.). In a neighborhood of U, the shock speed and the characteristic speed are positive, so that stationary shocks can be followed by -waves, only: using ϕ (U, a), the solution can begin with a stationary shock to a state U G, followed by using W (U ) as long as the shock speed is non-negative, i.e., (.8) U W (U ) : ρ ψ (U ). This is the curve W (U, a). Clearly, this curve crosses the curve C +. If the solution jumps by a stationary shock using ϕ (U, a) to a state U, then we know by Lemma. that U G in which -wave speeds are always negative so that -waves can not follow. In other words, W (U, a) is empty. On the other hand, from U, the construction can begin with a non-positive shock from U to some state U with ρ ψ (U ), followed by a stationary shock using either ϕ (U, a) to a state U with U G if u, U G if u,

13 B A C C D C C 8 NOZZLE WITH DISCONTINUOUS CROSS-SECTION L KRD FH9 NJOQI 9;: D DSKML FH9 :NJOQI? T D DHKML FH9 :NPOQI L DHKML FH9 : NJOQI?@ <>= CEDGFH9 :JI Figure.: U above C + or using ϕ (U, a) to a state U G, These two ways determine two composite curves W (U, a), W (U, a). When using ϕ (U, a) to attain a point U G, similarly, by virtue of (.6) the construction can be continued by using W (U ) as much as -shock speeds from U are non-negative. In this case, the composite wave set consists of three curves and a one-parameter family of solutions: (.9) W, (U, a) W (U, a) W (U, a) W (U, a) W (U, a) U W (U ), U between Ũ and Z(U ), where Ũ := (ψ (U ), ω (U, ψ (U ))) the point at which the -shock from U has zero speed. The construction is complete. Obviously, the curve W (U, a) is monotone, as a part of W (U ) for some U. Moreover, it is derived from the above construction that the two curves W (U, a), W (U, a) can be parameterized by ρ: (.) ρ (ϕ i (ρ, ω (U ; ρ); a), ω ((ρ, ω (U ; ρ)); ϕ i (ρ, ω (U ; ρ), a))), i =,, where ϕ i (ρ, ω (U ; ρ)) is defined by the implicit equation (.) Σ((ρ, ω (U ; ρ)), ϕ i (ρ, ω (U ; ρ)); a) =. We will show that parts of these two curves are monotone. These parts are sufficient to the construction of Riemann solutions in the next section. Lemma.. The u-component of the curve W, (U, a) is a decreasing function of ρ. The u-component of the curve W (U, a), as a function of ρ, is increasing for ρ < ρ and decreasing for ρ > ρ, where the point (ρ, ω (U, ρ )) G.

14 LEFLOCH AND THANH Proof. Without loss of generality we can assume that u >. For simplicity we may omit U in (.). Obviously, for both components u of the two curves we have (.) u (ρ) = ω κγ [( a ω ρ ) ] γ (ρ) ρ γ. γ au Taking the derivative with respect to ρ on both sides we find [( uu = ω ω κγ a ω ρ ) γ a aω a = ω ω κγ ( ϕ γ i (ω + ω ρ)ω ω ω ρ ω (ω + ω ρ)ω ω ω ρ ω ω ρ Combining together terms having ω of the last equality, we obtain ρ γ ). ρ γ ] (.) (u κγϕ γ i ) u u = ω ω + κγρ γ κγϕγ i (ω + ω ρ). ω ρ Observe that the first factor on the left-hand side of (.) (.) u (ρ) κγϕ i (ρ) γ > for i =, u (ρ) κγϕ i (ρ) γ < for i =. For the right-hand side we proceed as follows. For U on the curve W (U ) we have (.5) ω ω + κγργ κγϕγ i (ω + ω ρ) = ω (ω ω ρ ω κγϕγ i ) + κγ ρ (ργ ϕ γ i ) <, which is positive from U down to infinity for i =, but negative between U and U for i =. From now on we need only consider the case i = as the other case is similar. Therefore we just consider the part ρ > ρ in which (.6) (ρ, ω (ρ)) G. The right-hand side of (.) is estimated as (.7) ω ω + κγργ κγϕγ i (ω + ω ρ) < ω ω ω ρ + ω κγϕγ i (ω + ω /ρ ρ) ω ρ ω + ω ρ (ω ω ρ κγϕγ i ). The first factor on the right-hand side of (.7) is positive. We will show that the second factor is negative. Actually, (.8) ω κγϕ γ i = γ (u ω ) + ω κγρ γ := l(ρ). Taking the derivative of the function l(ρ) we obtain l (ρ) = (γ )(uu ω ω ) + ω ω κγ(γ )ρ γ,

15 NOZZLE WITH DISCONTINUOUS CROSS-SECTION 5 or (.9) γ l (ρ) = κω u ω [ ρ γ ρ γ ρ + γρ γ ( ρ ρ ) ] κγρ γ = κω (ρ ρ ) (u ω )ρ ( ρ γ ρ γ ρ ρ + γργ ρ ) κγρ γ κγργ ω (ρ ρ ) u ω κγρ γ <, ρ > ρ, and tends to as ρ +. The fact that the right-hand side of (.) takes a negative value at U and the inequality (.9) implies that the right-hand side of (.) is always negative. It is then derived from (.) that the function defined in (.) is dereasing for u <. It is similar to prove the case u >. The proof of Lemma. is complete. Lemma.. The ρ-component of the curve W (U, a) is increasing in ρ at least when (ρ, ω (ρ)) remains in G. Proof. Let the function u be defined as in (.). Taking the derivative in ρ of the identity u(ρ)ϕ (ρ) = a ω (ρ)ρ/a, we obtain or ϕ u + u ϕ = a (ω + ω ρ)/a, ϕ (ρ) = ϕ u /u + a (ω + ω ρ)/a. Replacing u /u from (.) in the last equality and after rearranging the terms, we obtain ( (.) ϕ (ρ) = ϕ ω + ω (ρ) ρ ω ρ As long as (ρ, ω (ρ)) belongs to G, it follows from (.) that (u κγϕ γ i ) u u = ω ω + κγρ γ κγϕγ i (ω + ω ρ) ω ρ ) u. u ω ω + ω/ρ κγϕγ i (ω + ω ρ) ω ρ = ω + ω ρ (ω κγϕ γ i ). ω ρ Thus, so that (.) yields ω + ω ρ ω ρ u u κγϕ γ i u ω > u κγϕγ u, i ϕ (ρ) >. This completes the proof of Lemma..

16 6 LEFLOCH AND THANH Corollary.. The curve W (U, a) can be parameterized by expressing the ρ-component as a function of the u-component, i.e., ρ = ρ(u). Moreover, this function is monotone decreasing, at least for (ρ, ω (ρ)) G. Proof. As seen by Lemma., the component u = u(ρ) of W (U, a) is a decreasing function of ρ. So, it admits in inverse ρ = u (u). Using this identity into the expression of the component ρ in (.), we get the component ρ = ρ(u). Lemmas. and. yield the desired monotonicity property.. The monotonicity criterion : Solutions with one stationary wave As seen in the previous section that the Riemann problem for (.) may admit up to a one-parameter family of solutions. This phenomenon can be avoided by requiring Riemann solutions to satisfy a monotone condition on the component a. Monotonicity Criterion. (a) Along any stationary curve W (U ), the cross-section area a is monotone as a function of ρ. (b) The total variation of the cross-section component of any Riemann solution must not exceed (and, therefore, is equal to) a l a r, where a l, a r are left-hand and right-hand cross-section levels. A similar criterion was used by Isaacson and Temple [9, ] and by Goatin and LeFloch [6]. In this section we will consider Riemann solutions containing one stationary wave only. Consequently, only the requirement (a) of the a-monotone criterion affects on the construction of solutions. The requirement (b) will be taken into account for solutions containing two stationary waves, which will be discussed in the next section. Under our monotonicity criterion, we will show that there exist two solutions containing one stationary wave at most. As before, for definiteness we assume in this section that (.) a l < a r. Moreover, the notation W i (U, U ) W j (U, U ), will be used when the i-wave connecting some state U to some state U is followed by a j-wave connecting U to U (here i, j =,, ). Lemma.. The Monononicity Criterion (a) is equivalent to saying that any stationary shock does not cross the boundary of strict hyperbolicity. In other words: (i) If U G G, then only the stationary shock based on the value ϕ (U, a) is allowed. (ii) If U G, then only the stationary shock using ϕ (U, a) is allowed. Proof. Taking the derivative with respect to ρ in the identity we get a (ω ρ) = (a u ρ ), (.) a(ρ)a (ρ)(ω ρ) + a (ω ρ)(ω ρ + ω ) =.

17 j k e U NOZZLE WITH DISCONTINUOUS CROSS-SECTION 7 To demonstrate the lemma, it is sufficient to show that the last factor of the second term in (.) remains of a constant sign whenever the curve W (U ) does not cross the boundary of strict hyperbolicity C ±. Indeed, assume for simplicity that u >, then ω (ρ)ρ + ω = κγργ ω + ω = ω κγρ γ ω, which remains of a constant sign if and only if W (U ) does not cross C ±. This completes the proof of Lemma.. ]_^ bdc eml f VZY g VW V X VR[ \ V Y VZY eiy f V W g n o e Y f \ V YPg [f V Xhg ]a` b Y Figure.: U r below or on C Let us denote by W i,a (U ) the part of W i (U, a), i =,, which is admissible for Monotonicity Criterion. In view of Lemma. we have (.) W,a (U ) = {U W (U, a), ρ ρ }, W,a (U ) = {U W (U, a), ρ ρ }, where ρ is the value at which the curve W (U ) intersects the boundary C.

18 w x u 8 LEFLOCH AND THANH Lemma.. When U describes the curve W (U ), U G from the boundary C to the horizontal axis {u = }, the critical value a min (U) decreases to zero. Given left-hand and right-hand states U l = (ρ l, u l, a l ), U r = (ρ r, u r, a r ), we now discuss the construction algorithm for the corresponding Riemann solution. We need some further notation. First, define the two curves C, C made of states reached by stationary jumps from points on C when the variable a jumps from a l to a r. Here we use ϕ (U(a = a l ), a r ) and ϕ (U(a = a l ), a r ), U C, respectively. Obviously, we have C G, C G. Lemma.. The curves C, C can be parameterized so that the u-component is decreasing as a function of the ρ-component. Second, it is convenient to introduce the two points at which the curve W (U l ) crosses the boundary of strict hyperbolicity: (.) {U ± } := W (U l ) C ±. In view of Proposition., for each U C, there exists a point Ũ G such that the -shock speed vanishes: λ (U, Ũ) =. Such states Ũ where U C form a curve denoted by C. ~ v psh y{z y p q v pmqh pt pv prs pzr ~ir pt ~ s pzr ƒ ~ r p s y s ym} y s y r Figure.: U r between or on C, C

19 Š NOZZLE WITH DISCONTINUOUS CROSS-SECTION 9 Construction. U r is below or on the curve C (Figure.). The solution may have a (right-to-left) jump from U r to U G C by ϕ (U r, a l ). In view of Proposition., there exists a point Ũ W (U ) at which the -shock speed vanishes. Then, define U by {U } = W (U l ) W (U ). The solution can be (.5) W (U l, U ) W (U, U ) W (U, U r ), which makes sense if and only if the -shock speed λ (U, U ) is non-positive. In other words, U l is below or on the curve W (Ũ). ĥ ˆ Š{ ŠŒ Z R ŠŽ Figure.: U l below C, U r above C Construction. U r is between or on the two curves C, C ( Figure.). Let {U } = W (U r ) C. Since U C, by definition, there exists a point U C which can be attained from U by using a stationary shock. Let {U } = W (U l ) W (U ).

20 «š LEFLOCH AND THANH The solution is then (.6) W (U l, U ) W (U, U ) W (U, U ) W (U, U r ), provided U is above or on the curve W (U l ). In other words, U l is below or on the curve W (U ). Construction. U l is below or on the curve C, and U r is above or on the curve C ( Figure.). Set {U } = W (U r ) {ρ = }, {U } = W (U l ) {ρ = }. The solution is then (.7) R (U l, U ) R (U, (O, a l )) W ((O, a l ), (O, a r )) R ((O, a r ), U ) R (U, U r ). šh š & š Ÿ š Ÿ š š ª šr šm {µ š h š š š š šœž š šœ ³ G± ² š ª ž š ª Figure.: U r between W (U ), W (U ) Construction. U l is below or on C +. At U, the solution can jump by a stationary wave using either ϕ (U, a r ) to U G, or ϕ (U, a r ) to a state U G. At U +, the solution can jump by a stationary wave using either ϕ (U +, a r ) to U G, or ϕ (U +, a r ) to a state U G. It is easy to see that U i is an endpoint of W i,a (U l, a r ), i =,.

21 Ì À    Š Ż Á º NOZZLE WITH DISCONTINUOUS CROSS-SECTION On the other hand, by virtue of Proposition., from any point U W,a (U l, a r ), the exists a point Ũ W (U) G such that the -shock speed from U to Ũ vanishes: λ (U, Ũ) =. Such states Ũ form a curve in G, denoted by L. On the curve W (U ) there exists a point Ũ G at which the -shock speed vanishes: λ (U, Ũ) =. This construction gives a (unique) solution in the following two cases: ŖÐ Â+È Ã Ä ¾ Ŗ» ¹ M½ Â Ç Ã» Ä Ã ½hÄ ÍÎ ÂÏ Ã ¹ Ä ÅMÆ ÍÎ Ã ¾ Ä ¼ œº ÇPÉ Ê È Ë Ç Ã ¼ Ä È Ã ¼ Ä Figure.5: U r above W (Ũ) (i) Assume that U r is above or on the curves L and W (U ), and is below or on the curve W (U ) ( Figure.). Set U 5 W (U r ) W i,a (U l, a r ), U 6 W (U 5 ) W (U l ). The solution is then (.) W (U l, U 6 ) W (U 6, U 5 ) W (U 5, U r ).

22 à á Û Û Ñ LEFLOCH AND THANH (ii) Assume that U r is above or on the curve W (Ũ) ( Figure.5). If W (U r ) W (U ) =, then the solution contains a component of empty density. Otherwise, set The solution is then {U 7 } = W (U r ) W (U ). (.) R (U l, U + ) W (U +, U ) W (U, U 7 ) W (U 7, U r ). ÛmÔ Ü Ò Ø Ý ÒÓ ÒZÖ ÒMØ Pâ ã ÔJä Ü Ò Ø Ý Ù{ß ÙÚ ÒÔ Ò Õ Ü Ò ÔÝ ÒR Û+Ö Ü Ò Ó Ý ÙMÞ Figure.6: U above W (U l ) Construction 5. U r is above C and is below C. This construction gives the same solutions as Constructions,. Let us denote by W,a (U r) the composite (backward) curve constisting of all the states U such that there exists some state U W (U r ) and the stationary shock from U using ϕ (U, a r ) admissible to the a-monotone criterion to U can be followed by the -wave from U to U r. We then define {U } = W,a (U r) C +, {U } = W,a (U r) C.

23 ì í î î î å NOZZLE WITH DISCONTINUOUS CROSS-SECTION æzç èpô õ ïpö è ðæ ë ñ î ïð æmëhñ æmë è ðæ è ñ ò{ù òó æé æ ærè ærê òmø ç ð æ é ñ Figure.7: U below W (U l ) Then, if U l is below or on the curve W (U ), the following construction gives a unique solution: (i) If U is above or on the curve W (U l ), then See Figure.6. W,a (U r) W (U l ) = {U }. By definition, the point U determines a point U W (U r ) to which the solution jumps by a stationary shock. The solution is then (.) W (U l, U ) W (U, U ) W (U, U r ). This solution however coincides with the one given by Construction. (ii) If U is below the curve W (U l ), then the point U determines a point U 5 W (U r ) to which the solution can use a stationary jump, and See Figure.7. The solution is then W (U l ) W (U ) = {U 6 }. (.) W (U l, U 6 ) R (U 6, U ) W (U, U 5 ) W (U 5, U r ), which coincides with the one given by Construction.

24 û ÿ ú LEFLOCH AND THANH +ý ü þ üzý ü þ +ýüzý ü þ ü üzý üzý ý ü þ ü ü ü ü ü üœÿ ü ü Figure.8: U r between W (U ) and W (U ) Construction 6. U l G. From U l, a stationary jump using ϕ (U l, a r ) attains a state U, the solution can be continued by using W (U ) as long as the -shock speed from U is non-negative, i.e., until a state Ũ at which the -shock speed from U vanishes. On the other hand, the solution can jump by a non-positive shock from U l to any state U below or at a state Ũl at which the -shock speed from U l vanishes. The solution then can be continued by a stationary jump using ϕ (U, a r ) where U W (U l ). Let us denote U, U the states reached by a stationary jump from Ũl, U, respectively, using ϕ (Ũl, a r ), ϕ (U, a r ), respectively. (i) Assume that U r is above or on the curve W (U ), and is below or on the curve W (U ) (Figure.8). Then W,a (U l) W (U r ) = {U }. By definition, the point U determines a point U 5 belonging to W (U l ) corresponding to a stationary jump. The solution is then (.) S (U l, U 5 ) W (U 5, U ) W (U, U r ), provided the -shock speed λ (U, U r ) is non-negative, i.e., U r is above or on a curve L - the set of all states at which the -shock speed from any state U on the part {u } of the curve W,a (U l) vanishes.

25 + ' ' / NOZZLE WITH DISCONTINUOUS CROSS-SECTION 5, /! )5 /6"! $ -.! ) /!#" ' -.!#" /! (! $ :; / "! "! *! $! (!#"!&% (7 8 "9 (! $ : % Figure.9: U r above W (Ũ) (ii) Now, let U r be above or on the curve W (Ũ) (Figure.9). If (.5) W (U r ) W (U ) = {U 6 }. The solution is then (.6) W (U l, U ) W (U, U 6 ) W (U 6, U r ). If (.5) is not satisfied, i.e., W (U ) {ρ = } = {U 7 }, W (U r ) {ρ = } = {U 8 }, so that U 8 is above U 7, the solution then contains a component of empty density.

26 F E I < 6 LEFLOCH AND THANH 5. Solutions with two stationary waves GH I AJ = D5K GH I ALJ = CLK MN IO J = A K = > = D = O =#A? = > = B = C M A IPA J K BJ = >K M Q Figure 5.: U r G In this section we will complete the description of Riemann solutions by considering solutions that contain two stationary waves. Under the Monotonicity Criterion, this happens when a stationary shock jumps from the level a = a l to an intermediate level a between a l and a r, and then we use a k-shock with zero speed which keeps constant the level a, k =,, then we finally use another stationary shock jumping from the level a to a r. Hence, there are only two possibilities: (i) U r belongs to G and the shock with zero speed is a -shock, (ii) U l belongs to G and the shock with zero speed is a -shock. We are going to discuss these two situations. Construction 7. U r G (Figure 5.). Any a (a l, a r ) determines a state U W (U r ) which can be attained by a stationary jump using ϕ (U r, a). In view of Proposition., there exists a point U W (U ) such that the -shock speed λ (U, U ) =. The state U determines a state U W (U ) attainable by a stationary jump using ϕ (U (a), a l ). The curve W (U l ) passing through U will determine the Riemann solution: (5.) W (U l, U ) W (U, U ) S (U, U ) W (U, U r ). In order for this construction to make sense, the following requirements must be imposed (5.) { a amin (U r ), a l a min (U ).

27 Z Y R NOZZLE WITH DISCONTINUOUS CROSS-SECTION 7 ]PV`S&Vb S#V S ^ S a S T ]_^`S ab [\ S U ]^`S XLb [\ ]gw`s T b S X S W df ]6V`S Tcb ]^`S Ub d e Figure 5.: U l G Observe that a min (U r ) < a l (< a) if and only if U r is below the curve C so that the first line in (5.) is obvious. The condition (5.) can be seen as a constraint on the contracting duct. Construction 8. U l G (Figure 5.). Any a (a l, a r ) determines a state U W (U l ). It is derived from Proposition. that there exists a point U W (U ) such that the -shock speed λ (U, U ) =. The state U determines a state U W (U ) attainable by a stationary jump using ϕ (U (a), a r ). The curve W (U r ) passing through U will determine the Riemann solution: (5.) W (U l, U ) S (U, U ) W (U, U ) W (U, U r ). There is no constraint on expanding duct.

28 8 LEFLOCH AND THANH 6. Numerical results In this section we plot the solution of the Riemann solution for various range of values of the data. Construction. (See Figure., Section ) Figure C.: S R Z Figure C.: S S Z Figure C.: The solution is a -shock from U l = (ρ l, u l, a l ) to U followed by a -rarefaction wave from U to U, followed by a stationary wave from U to U r = (ρ r, u r, a r ). Figure C.: The solution is a -shock from U l = (ρ l, u l, a l ) to U followed by a -shock from U to U, followed by a stationary wave from U to U r = (ρ r, u r, a r ). Construction. (See Figure., Section ) Figure C.: R R Z R Figure C.: R R Z S Figure C.: The solution is a -rarefaction wave from U l = (ρ l, u l, a l ) to U followed by a -rarefaction wave from U to U, followed by a stationary wave from U to U, followed by a -rarefaction wave from U to U r = (ρ r, u r, a r ).

29 NOZZLE WITH DISCONTINUOUS CROSS-SECTION Figure C.: S R Z R Figure C.: S R Z S Figure C.: The solution is a -rarefaction wave from U l = (ρ l, u l, a l ) to U followed by a -rarefaction wave from U to U, followed by a stationary wave from U to U, followed by a -shock from U to U r = (ρ r, u r, a r ). Figure C.: The solution is a -shock wave from U l = (ρ l, u l, a l ) to U followed by a -rarefaction wave from U to U, followed by a stationary wave from U to U, followed by a -rarefaction wave from U to U r = (ρ r, u r, a r ). Figure C.: The solution is a -shock wave from U l = (ρ l, u l, a l ) to U followed by a -rarefaction wave from U to U, followed by a stationary wave from U to U, followed by a -shock from U to U r = (ρ r, u r, a r ). Construction. (See Figure., Section ) Figure C.: R R Z R R Figure C.: The solution is a -rarefaction wave from U l = (ρ l, u l, a l ) to U followed by a rarefaction wave with zero density, i.e. in empty, from U to O at which it suffers a shift on the component a,

30 LEFLOCH AND THANH followed by a rarefaction wave with zero density from O to U, followed by a -rarefaction wave from U to U r = (ρ r, u r, a r ). Construction. (See Figure., Section ) Figure C.: R Z R Figure C.: R Z S Figure C.: The solution is a -rarefaction wave from U l = (ρ l, u l, a l ) to U 6 followed by a stationary wave from U 6 to U 5, followed by a -rarefaction wave from U 5 to U r = (ρ r, u r, a r ). Figure C.: The solution is a -rarefaction wave from U l = (ρ l, u l, a l ) to U 6 followed by a stationary wave from U 6 to U 5, followed by a -shock from U 5 to U r = (ρ r, u r, a r ) Figure C.: S Z R Figure C.: S Z S Figure C.: The solution is a -shock wave from U l = (ρ l, u l, a l ) to U 6 followed by a stationary wave from U 6 to U 5, followed by a -rarefaction wave from U 5 to U r = (ρ r, u r, a r ). Figure C.: The solution is a -shock wave from U l = (ρ l, u l, a l ) to U 6 followed by a stationary wave from U 6 to U 5, followed by a -shock from U 5 to U r = (ρ r, u r, a r ). Construction 5. (See Figure.5, Section ) Figure C5.: The solution is a -rarefaction wave from U l = (ρ l, u l, a l ) to U + followed by a stationary wave from U + to U, followed by a -rarefaction wave from U to U 7, followed by a -rarefaction wave from U 7 to U r = (ρ r, u r, a r ).

31 NOZZLE WITH DISCONTINUOUS CROSS-SECTION Figure C5.: R Z R R Figure C5.: R Z R S Figure C5.: The solution is a -rarefaction wave from U l = (ρ l, u l, a l ) to U + followed by a stationary wave from U + to U, followed by a -rarefaction wave from U to U 7, followed by a -shock wave from U 7 to U r = (ρ r, u r, a r )..5.5 Figure C5.: R Z S R Figure C5.: The solution is a -rarefaction wave from U l = (ρ l, u l, a l ) to U + followed by a stationary wave from U + to U, followed by a -shock wave from U to U 7, followed by a -rarefaction wave from U 7 to U r = (ρ r, u r, a r ). Construction 6. (See Figure.8, Section ) Figure C6.: The solution is a -shock wave from U l = (ρ l, u l, a l ) to U 5 followed by a stationary wave from U 5 to U, followed by a -rarefaction wave from U to U r = (ρ r, u r, a r ). Figure C6.: The solution is a -shock wave from U l = (ρ l, u l, a l ) to U 5 followed by a stationary wave from U 5 to U, followed by a -shock from U to U r = (ρ r, u r, a r ). Construction 7. (See Figure.9, Section )

32 LEFLOCH AND THANH Figure C6.: S Z R Figure C6.: S Z S Figure C7.: Z R R Figure C7.: Z R S Figure C7.: The solution is a stationary shock from U l = (ρ l, u l, a l ) to U followed by a -rarefaction wave stationary wave from U to U 6, followed by a -rarefaction wave from U 6 to U r = (ρ r, u r, a r ). Figure C7.: The solution is a stationary shock from U l = (ρ l, u l, a l ) to U followed by a -rarefaction wave stationary wave from U to U 6, followed by a -shock wave from U 6 to U r = (ρ r, u r, a r ). Figure C7.: The solution is a stationary wave U l = (ρ l, u l, a l ) to U, followed by a -shock wave from U to U 6, followed by a -rarefaction wave from U 6 to U r = (ρ r, u r, a r ). References [] Botchorishvili R., Perthame B., and Vasseur A., Equilibrium schemes for scalar conservation laws with stiff sources, INRIA Preprint,. [] Bouchut F., An introduction to finite volume methods for hyperbolic systems of conservation laws with source, INRIA Rocquencourt report,. [] Chen G.-Q. and Glimm J., Global solutions to the compressible Euler equations with geometrical structure, Commun. Math. Phys. 8 (996), 5 9. [] Courant R. and Friedrichs K.O., Supersonic flow and shock waves, John Wiley, New York, 98. [5] Dal Maso G., LeFloch P.G., and Murat F., Definition and weak stability of nonconservative products,

33 NOZZLE WITH DISCONTINUOUS CROSS-SECTION Figure C7.: Z S R J. Math. Pures Appl. 7 (995), 8 58 [6] Goatin P. and LeFloch P.G., The Riemann problem for a class of resonant nonlinear systems of balance laws, Preprint. [7] Gosse L., A well-balanced scheme using non-conservative products designed for hyperbolic conservation laws with source-terms, Math. Model Methods Appli. Sc. (), [8] Greenberg J.M. and Leroux A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. (996), -6. [9] Isaacson E. and Temple B., Convergence of the Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (995), [] Isaacson E. and Temple B., Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math. 5 (99), [] Keyfitz B.L., A survey of nonstrictly hyperbolic conservation laws. Nonlinear hyperbolic problems (St. Etienne, 986), 5 6, Lecture Notes in Math., 7, Springer, Berlin, 987. [] Lax P.D., Shock wave and entropy, in Contributions to Functional Analysis, ed. E.A. Zarantonello, Academic Press, New York, 97, 6-6. [] LeFloch P.G., Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Math. and its Appl., Minneapolis, Preprint # 59, 989. [] LeFloch P.G., Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zürich, Birkäuser,. [5] LeFloch P.G. and Thanh M.D., Properties of Rankine-Hugoniot curves for Van der Waals fluid flows, Japan J. Indus. Applied Math. (). [6] Liu T.-P., Nonlinear stability and instability of transonic flows through a nozzle, Commun. Math. Phys 8 (98), 6. [7] Liu T.-P., Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 8 (987), [8] Marchesin D. and Paes-Leme P.J., A Riemann problem in gas dynamics with bifurcation. Hyperbolic partial differential equations III, Comput. Math. Appl. (Part A) (986), 55.