DSpace at VNU: Numerical approximation for a Baer-Nunziato model of two-phase flows

DSpace at VNU: Numerical approximation for a Baer-Nunziato model of two-phase flows tài liệu, giáo án, bài giảng , luận...

Applied Numerical Mathematics 61 (2011) 702–721 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum Numerical approximation for a Baer–Nunziato model of two-phase flows Mai Duc Thanh a,∗ , Dietmar Kröner b , Nguyen Thanh Nam c a b c Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam Institute of Applied Mathematics, University of Freiburg, Hermann-Herder Str 10, 79104 Freiburg, Germany National Key Laboratory of Digital Control and System Engineering, Block C6, 268 Ly Thuong Kiet street, Ward 14, District 10, Ho Chi Minh City, Viet Nam a r t i c l e i n f o Article history: Received February 2009 Received in revised form 13 January 2011 Accepted 13 January 2011 Available online 18 January 2011 Keywords: Two-phase flow Conservation law Source term Numerical approximation Lax–Friedrichs Well-balanced scheme a b s t r a c t We present a well-balanced numerical scheme for approximating the solution of the Baer–Nunziato model of two-phase flows by balancing the source terms and discretizing the compaction dynamics equation First, the system is transformed into a new one of three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation law of the mass in the solid phase and the conservation law of the momentum of the mixture, and the compaction dynamic equation is considered as the third subsystem In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states The second subsystem is of conservative form and thus can be numerically treated in a standard way For the third subsystem, the fact that the solid velocity is constant across the solid contact suggests us to compose the technique of the Engquist–Osher scheme We show that our scheme is capable of capturing exactly equilibrium states Moreover, numerical tests show the convergence of approximate solutions to the exact solution © 2011 IMACS Published by Elsevier B.V All rights reserved Introduction We are interested in numerical approximations for the solutions of the Baer–Nunziato (BN) model of two-phase flows which was introduced by Baer and Nunziato [4] for the study of the deflagration-to-detonation transition (DDT) in granular explosives This two-phase treatment of the explosive as a mixture is mathematically formulated in terms of variables for its two separate constituents: a granular solid phase and a separate combustion product gas phase The mixture is assumed to be immiscible but the two phases are not in equilibrium with each other Each phase is identifiable and characterized by a separate equation of state (EOS) The volume fraction of each phase is a dependent variable required to specify the state of the mixture Each phase satisfies the balance laws of mass, momentum and energy Interactions between two phases are described by source terms for the exchange of mass, momentum and energy between the phases The model is enclosed by the compaction dynamics equation which describes the evolution of the volume fraction variable This equation indicates the way in which microstructural forces at the interphases act to derive the volume fraction toward equilibrium states See [8] for a crucial review and the references therein for the modeling, analysis and numerical simulation of DDT in porous energetic materials The governing equations of the BN model are in accordance with the general framework [11] which formulates the mathematical derivation of multi-phase flow models See also [29,12], etc., for the modeling of two-pressure two-phase flows In this paper, we consider the BN model in the nonreactive and isentropic case Furthermore, the interfacial pressure is assumed to be equal to the pressure in the gas phase, and the mass and the residual momentum exchanges between phases * Corresponding author E-mail addresses: mdthanh@hcmiu.edu.vn (M.D Thanh), dietmar@mathematik.uni-freiburg.de (D Krưner), thanhnam@dcselab.edu.vn (N.T Nam) 0168-9274/$30.00 © 2011 IMACS Published by Elsevier B.V All rights reserved doi:10.1016/j.apnum.2011.01.004 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 703 are neglected Precisely, the model is described by a system of four equations characterizing the conservation of mass and the balance of momentum in each phase: ∂t (α g ρ g ) + ∂x (α g ρ g u g ) = 0, ∂t (α g ρ g u g ) + ∂x α g ρ g u 2g + p g = p g ∂x α g , ∂t (αs ρs ) + ∂x (αs ρs u s ) = 0, ∂t (αs ρs u s ) + ∂x αs ρs u 2s + p s = − p g ∂x α g , (1.1) together with the compaction dynamics equation ∂t α g + u s ∂x α g = 0, x ∈ R, t > (1.2) Throughout, we use the subscripts g and s to indicate the quantities in the gas phase and in the solid phase, respectively The notations αk , ρk , uk , pk , k = g , s, respectively, stand for the volume fraction, density, velocity, and pressure in the k-phase, k = g , s The volume fractions satisfy α s + α g = (1.3) Each phase has an equation of state of the form γ pk = κk ρk k , κk > 0, < γk < 5/3, k = s, g (1.4) The system (1.1)–(1.2) has the form of nonconservative systems of balance laws In nonconservative systems, there are often a part of conservative terms and a part of nonconservative terms Without nonconservative terms, the systems can be well dealt with by the standard theory of hyperbolic systems of conservations laws Nonconservative terms such as the terms p g ∂x α g , u s ∂x α g in the above model or source terms in other models (see [24,34,36] for instance) in general involve the product of a discontinuous function and the (partial) derivative of another quantity that might be discontinuous as well Mathematical formulation of nonconservative systems of balance laws was introduced in [10] The study of the impact of nonconservative terms plays a key role in nonconservative systems and, in particular, in multi-phase flows models In general, nonconservative terms may cause the ill-posedness for the boundary/initial-value problem, as there can be multiple solutions, see [26,34], for example There have been many contributions for the study of wave structures and the Riemann problem for various models of nonconservative systems An early research for the model of a fluid in a nozzle with discontinuous cross-section was carried out in [24] This was followed by a sequence of papers [26,34] for the same model, [27,36] for shallow water equations, and [13] for the case of a general system In [34], the uniqueness of the Riemann solutions of the model of a fluid in a nozzle with piece-wise cross-section has been established In [2,32], exact Riemann solutions of the BN model were constructed See also the references therein for related works Numerically, the nonconservative terms often cause lots of inconveniences in approximating physical solutions of the system This has been seen even in the case of a single conservation law with a source term (see [6] for example) The discretization of nonconservative terms therefore has been an attractive topic for many years Basically, a good numerical method for a nonconservative system should give a good approximation of the exact solution in the typical case where the system is stable and stationary (independent of time) whenever such a situation exists This case corresponds to the equilibrium states which can be obtained using stationary waves This has motivated, particularly, the study of well-balanced schemes for nonconservative systems that can capture equilibrium states, see [6,21,36,18,19,37] for equilibrium sate capturing schemes for various models Many nonconservative systems, such as the ones we just talked about above, possess the characteristic property that: all the eigenvalues of the Jacobian matrix are real and given in explicit forms, and that the characteristic fields may coincide on certain surfaces of the phase domain This is a very interesting phenomenon in the sense that on one hand the nonconservativeness poses challenging problems, on the other hand the fact that all the characteristic fields are real and explicit raises the hope for the study in the framework of hyperbolic systems Observe that many one-pressure two-fluid models, where pressures on both phases are assumed to be equilibrium with each other, not possess this property, see [33] As seen later on, the system (1.1)–(1.2) is hyperbolic as all the characteristic fields are real and explicitly given Moreover, it is not strictly hyperbolic as some characteristic fields may coincide on certain surfaces of the phase domain Our goal in this paper is to provide a reliable numerical method for approximating the solution of the system (1.1)–(1.2) that can capture equilibrium states First, to reduce the nonconservative terms, we transform the system into a new one which is combined from three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation laws of the mixture, and the compaction dynamic equation represents the third subsystem Observe that each of these three subsystems is different from each other in the nature and thus will be treated separately and differently In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states This explain why our method can capture exactly equilibrium states The second subsystem is numerically treated in a usual way as it is conservative For the third subsystem, we observe that the solid velocity is constant across the solid contact and that the compaction dynamics equation becomes conservative in the region where the velocity of the solid phase is constant This suggests us to employ the technique of the Engquist–Osher scheme to treat the 704 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 compaction dynamics equation We show that our scheme capture exactly the equilibrium states and numerical tests provide us with reasonable approximations of the exact Riemann solutions of the model There have been many contributions devoted to the discretization of the nonconservative terms in nonconservative systems of balance laws In the case of a single conservation law with a source term, numerical well-balanced schemes were presented in [14,15,6,5,3] A well-balanced scheme for the model of fluid flows in a nozzle with variable cross-section was proposed in [21] and its properties are studied in [22] Well-balanced schemes for one-dimensional shallow water equations were constructed in [36,9] In [31] the authors propose to take into account the nonconservative terms using the free streaming physical condition with uniform velocity and pressure profiles See also [7,23,30] and the references therein for related works A well-balanced scheme for a one-pressure model of two-phase flows where one phase is compressible, the other phase is incompressible was constructed in [35], where the impact of stationary waves is required to take place on the full system In [28], the author takes the comparison of Roe-type methods for solving a two-fluid model with and without pressure relaxation Recently, a relaxation and numerical approximation were presented in [1], and a hybrid scheme was presented in [20] for the Baer–Nunziato model The outline of this paper is as follows In Section we provide basic properties of the system (1.1)–(1.2) Section deals with stationary waves In Section 4, we construct the numerical method for the BN model Section is devoted to numerical tests Preliminaries 2.1 Hyperbolicity For smooth solutions, the system (1.1)–(1.2) is equivalent to the following system ∂t ρ g + u g ∂x ρ g + ρ g ∂x u g = 0, ∂t u g + h g (ρ g )∂x ρ g + u g ∂x u g = 0, ∂t ρs + u s ∂x ρs + ρs ∂x u s = 0, ∂t u s + h s (ρs )∂x ρs + u s ∂x u s + ∂t α g + u s ∂x α g = 0, p g − ps (1 − α g )ρs ∂ x α g = 0, x ∈ R, t > 0, (2.1) where hk is the specific enthalpy of the k-phase: hk (ρ ) = pk (ρ ) ρ , k = s, g From (2.1), choosing the dependent variable U = (ρ g , u g , ρs , u s , α g ), we can rewrite the system (1.1)–(1.2) as a system of balance laws in nonconservative form as U t + A ( U ) U x = 0, (2.2) where ⎛ ug ρg 0 0 0 us ρs h s (ρs ) us p g −ps ( −α g ) ρ s 0 us ⎜ h g (ρ g ) u g ⎜ A (U ) = ⎜ ⎜ ⎝ 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ The characteristic equation is given by (u s − λ) (u g − λ)2 − p g (u s − λ)2 − p s = 0, which admits five roots as λ1 ( U ) = u g − pg, λ3 ( U ) = u s − ps , λ2 ( U ) = u g + λ4 ( U ) = u s + pg, ps, λ5 ( U ) = u s Thus, the system is hyperbolic and the corresponding right eigenvectors can be chosen as (2.3) M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 −2 p g (ρ g ) r (U ) = T p g (ρ g )ρ g + 2p g (ρ g ) p g (ρ g ) r (U ) = 705 ρ g , − p g , 0, 0, , T p g (ρ g )ρ g + 2p g (ρ g ) ρ g , p g , 0, 0, , −2 p s (ρs ) T 0, 0, ρ s , − p s , , p s (ρs )ρs + 2p s (ρs ) r (U ) = p s (ρs ) r (U ) = p s (ρs )ρs + 2p s (ρs ) 0, 0, ρ s , p s (ρs ) − p g (ρ g ) r ( U ) = 0, 0, (1 − α g )ρs T ps , , T , 0, h s (ρs ) (2.4) It is not difficult to verify that D λ i ( U ) · r i ( U ) = 1, i = 1, 2, 3, 4, D λ5 ( U ) · r ( U ) = , (2.5) so that the first, second, third, fourth characteristic fields (λi (U ), r i (U )), i = 1, 2, 3, 4, are genuinely nonlinear, while the fifth characteristic field (λ5 (U ), r5 (U )) is linearly degenerate Observe that the characteristic fields (λ1 (U ), r1 (U )) and (λ2 (U ), r2 (U )) may coincide with any remaining characteristic fields (λ3 (U ), r3 (U )), (λ4 (U ), r4 (U )) and (λ5 (U ), r5 (U )) on certain surfaces Thus, the system is not strictly hyperbolic in the whole domain 2.2 Rarefaction waves Next, let us look for rarefaction waves of the system (2.2), i.e., the continuous piecewise-smooth self-similar solutions of the form x U (x, t ) = V (ξ ), ξ = , t > 0, x ∈ R t Substituting this into (2.2), we can see that rarefaction waves are solutions of the following initial-value problem for ordinary differential equations dV (ξ ) dξ = r i V (ξ ) , V λi U λ i U , i = 1, 2, 3, 4, ξ = U (2.6) For the first characteristic field, (2.6) yields dρ g (ξ ) dξ du g (ξ ) dξ dρs (ξ ) dξ = = = −2 p g (ρ g ) p g (ρ g )ρ g + 2p g (ρ g ) ρ g (ξ ) < 0, p g (ρ g ) p g (ρ g )ρ g + 2p g (ρ g ) du s (ξ ) dξ = dα g (ξ ) dξ p g (ξ ) > 0, = This implies that ρs , u s , α g are constant through 1-rarefaction waves, with respect to ξ Moreover, u g can be resolved from ρ g by du g dρ g = − p g (ρ g ) ρg , (2.7) ρ g is decreasing with respect to ξ and u g is increasing (2.8) which determines the curve R1 (U ) consisting of all right-hand states that can be connected to the left-hand state U using 1-rarefaction waves The trajectory u g = u g (ρ g ) of (2.8) starting at U is given by 706 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 ρg R1 U : ug = u 0g p g ( y) − y dy , ρ g0 , ρg (2.9) ρ g0 since the characteristic speed must be increasing through a rarefaction fan Argue similarly, we can see that ρs , u s , α g are constant through 2-rarefaction waves, and the second rarefaction curve R2 (U ) starting from U is given by ρg R2 U : ug = u 0g p g ( y) + y dy , ρ g0 ρg (2.10) ρ g0 Through 3-rarefaction fans, ρ g , u g , α g are constant, and the third rarefaction curve R3 (U ) starting from U is given by ρs R3 U : us = u 0s − ps ( y) y ρs0 ρs dy , (2.11) ρs0 Through 4-rarefaction fans, ρ g , u g , α g are constant, and the fourth rarefaction curve R4 (U ) starting from U is given by ρs R4 U : us = u 0s + ps ( y) y ρs0 ρs dy , (2.12) ρs0 2.3 Shock waves and contact discontinuities It has been shown that, see [2,32] for example, along a discontinuity: (i) either the volume fractions are constant; (ii) or u s ≡ constant and the discontinuity propagates with a constant speed λ = u s In the case (i), the system then becomes two independent subsystems of isentropic gas dynamics equations in each phase Given a left-hand state U , the shock curves Si (U ), i = 1, 2, 3, consisting of all right-hand states U that can be connected to U by a Lax shock are given by: S1,3 (U ): uk = uk0 − κk S2,4 (U ): uk = uk0 − κk ρk0 ρk0 − − ρk ρk γ ρkγ − ρk0 γ ρkγ − ρk0 1/2 , ρk > ρk0 , , ρk < ρk0 , k = g , s 1/2 (2.13) The four wave curves in nonlinear characteristic families are then defined by: Wi U := Ri U ∪ Si U , i = 1, 2, 3, It is easy to see that along the curves W1 (U ), W3 (U ), the velocity is a monotone decreasing function of the density; along the curves W2 (U ), W4 (U ), the velocity is a monotone increasing function of the density In the case (ii), the discontinuity satisfies the following jump relations α g ρ g ( u g − u s ) = 0, u 2g − u g u s + h g = 0, α g ρ g ( u g − u s ) u g + α g p g + α s p s = (2.14) Thus, the left-hand state U and right-hand state U of a contact discontinuity associated with the characteristic speed λ5 = u s ≡ constant satisfy α g ρ g (u g − u s ) = α g0 ρ g0 (u g0 − u s ) := m, (u g − u s )2 + 2h g = (u g0 − u s )2 + 2h g0 , mu g + α g p g + αs p s = mu g0 + α g0 p g0 + αs0 p s0 (2.15) M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 707 The above argument leads to defining elementary waves of the system (1.1)–(1.2), which make up solutions of the Riemann problem Definition 2.1 (Elementary waves) The elementary waves of the system (1.1)–(1.2) are (i) Rarefaction waves these waves are continuous piecewise-smooth self-similar solutions of the system (1.1)–(1.2); (ii) Lax shocks in each phase where the system becomes the usual isentropic gas dynamics equations; (iii) Contact discontinuities associated with the linearly degenerate characteristic field λ5 = u s where the left-hand and right-hand states are defined by (2.15) Stationary contacts in the gas phase As observed earlier, nonconservative terms often cause lots of inconveniences for numerical approximations To reduce the number nonconservative terms, we add up the two equations of balance of momentum to get the conservation of momentum of the total in place of the equation of balance of momentum for the liquid phase So we get three sets of equations: – governing equations in the gas phase: ∂t (α g ρ g ) + ∂x (α g ρ g u g ) = 0, ∂t (α g ρ g u g ) + ∂x α g ρ g u 2g + p g = p g ∂x α g , (3.1) – “composite” conservation laws: ∂t (αs ρs ) + ∂x (αs ρs u s ) = 0, ∂t (αs ρs u s + α g ρ g u g ) + ∂x αs ρs u 2s + p s + α g ρ g u 2g + p g = 0, (3.2) – compaction dynamics equation: ∂t α g + u s ∂x α g = (3.3) Set the dependent conservative variable V = (α g ρ g , α g ρ g u g , αs ρs , α g ρ g u g + αs ρs u s ) T , the flux α g ρ g u g , α g ρ g u 2g + p g , αs ρs u s , αs ρs u 2s + p s + α g ρ g u 2g + p g f (V ) = T , and the source S ( V ) = (0, p g ∂x α g , 0, 0) T We can see that a unique source appears only in the second component Thus, we can rewrite the system (3.1)–(3.2) as a system of conservation laws with a single source term ∂t V (x, t ) + ∂x f V (x, t ) = S V (x, t ) , x ∈ R, t > (3.4) Observe that the system (3.4) is under-determined as the number of unknowns is larger than the number of equations We still need the compaction dynamics equation (3.3) for the closure of the system We need to study stationary contacts of the system (3.4) which takes into account the nonconservative terms Therefore, we need only to consider the governing equations (3.1) of the gas phase To simplify the expressions, in the rest of this section, we omit the subscript g in the gas phase Motivated by our earlier works [26,21], we look for stationary contacts that are the limit of stationary smooth solutions of (3.1) A stationary smooth solution U of (3.1) is a time-independent smooth solution so that it satisfies the following ordinary differential equations (αρ u ) = 0, u2 +h = 0, where (.) = d/dx and h (ρ ) = p (ρ )/ρ , or h(ρ ) = κγ γ −1 ρ γ −1 Arguing similarly as in [26,21], we obtain the following conclusion (3.5) 708 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 Lemma 3.1 The left-hand and right-hand states U ± of a stationary contact for (3.1) satisfy [αρ u ] = 0, u2 + h = 0, (3.6) where [αρ u ] := α + ρ + u + − α − ρ − u − , and so on, denotes the difference of the corresponding value αρ u between the right-hand and left-hand states of the stationary contact From Lemma 3.1, we deduce that a stationary wave of (3.1) from a given state U = (α0 , ρ0 , u ) to some state U = (α , ρ , u ) must satisfy the relations αρ u = α0 ρ0 u , u2 + h(ρ ) = u 20 + h(ρ0 ) (3.7) It is derived from (3.7) that the density is a root of the nonlinear algebraic equation F (U , ρ , α ) := sgn(u ) u 20 − 2κγ 1/2 ρ γ −1 − ρ0γ −1 γ −1 ρ− α0 u ρ0 = α (3.8) To find zeros of the function F (U , ρ , α ), we need to investigate its properties Observe that the function F (U , ρ , α ) is well-defined whenever u 20 − 2κγ γ −1 ρ γ −1 − ρ0γ −1 0, or ρ ρ¯ (U ) := γ −1 γ −1 u + ρ0 2κγ γ −1 We have γ −1 2κγ γ −1 − ρ ) − κγρ γ −1 ∂ F (U , ρ ; α ) u − γ −1 (ρ = γ −1 1/2 2κγ ∂ρ u2 − (ρ γ −1 − ρ ) γ −1 Assume, for definitiveness, that u > The last expression yields ∂ F (U , ρ ; α ) > 0, ∂ρ ∂ F (U , ρ ; α ) < 0, ∂ρ ρ < ρmax (ρ0 , u ), ρ > ρmax (ρ0 , u ), where γ −1 ρmax (ρ0 , u ) := u 20 + ρ γ −1 κγ (γ + 1) γ +1 γ −1 (3.9) Since F (U , ρ = 0, a) = F (U , ρ = ρ¯ , a) = − the function α0 u ρ0 < 0, α ρ → F (U , ρ ; α ) admits a root if and only if the maximum value is nonnegative: F (U , ρ = ρmax , α ) 0, or, equivalently, α αmin (U ) := α0 ρ0 |u | √ γ +1 (3.10) κγ ρmax (ρ0 , u ) Similar argument can be made for u < It will be convenient to define in the (ρ , u )-plan the following sets, referred to as the “lower region” G , the “middle region” G , and the “upper region” G , and the “boundary” C , as M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 709 Fig The regions G , G , G and the boundary C in the (ρ , u )-plane G := (ρ , u ): u < − p (ρ ) , G := (ρ , u ): |u | < G := (ρ , u ): u > p (ρ ) , p (ρ ) , C := (ρ , u ): u = ± p (ρ ) , (3.11) see Fig The above argument leads us to the following results Lemma 3.2 Given U = (α0 , ρ0 , u ) and α The function F (U , ρ , α ) in (3.8) admits a zero if and only if a this case, F (U , ρ , α ) admits two distinct zeros, denoted by ρ = ϕ1 (U , α ), ρ = ϕ2 (U , α ) such that ϕ1 (U , α ) ρmax (U ) ϕ2 (U , α ) αmin (U ) In (3.12) the equality in (3.12) holds only if α = αmin (U ) Lemma 3.3 (i) Let ρmax (ρ0 , u ) be defined as (3.9) Then, it holds that ρmax (ρ0 , u ) < ρ0 , (ρ0 , u ) ∈ G , ρmax (ρ0 , u ) > ρ0 , (ρ0 , u ) ∈ G ∪ G , ρmax (ρ0 , u ) = ρ0 , (ρ0 , u ) ∈ C (3.13) (ii) Given U = (α0 , ρ0 , u ) and α Let u be defined by (3.7) and let ϕi (U , α ), i = 1, 2, be defined in Lemma 3.2 The state (ϕ1 (U , α ), u ) ∈ G if u < 0, and the state (ϕ1 (U , α ), u ) ∈ G if u > 0; the state (ϕ2 (U , α ), u ) ∈ G Moreover, ρmax (ρ0 , u ), u ∈ C (3.14) In addition, we have – If α > α0 , then ϕ1 (U , α ) < ρ0 < ϕ2 (U , α ) (3.15) – If α < α0 , then ρ0 < ϕ1 (U , α ) for (ρ0 , u ) ∈ G ∪ G , ρ0 > ϕ2 (U , α ) for (ρ0 , u ) ∈ G (3.16) (iii) Given U = (α , ρ , u ) and let αmin (U ) be defined as in (3.10) The following conclusions hold αmin (U ) < α , (ρ , u ) ∈ G i , i = 1, 2, 3, αmin (U ) = α , (ρ , u ) ∈ C , αmin (U ) = 0, ρ = or u = (3.17) 710 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 Proof Most of the proof was available in [26] However, for completeness, we will show the steps Assume for simplicity that u > Define g (U , ρ ) = u 20 − 2κγ ρ γ −1 − ρ0γ −1 − κγρ γ −1 γ −1 (3.18) Then, a straightforward calculation gives g U , ρmax (U ) = 0, which proves (3.14) On the other hand, since dg (U , ρ ) dρ and that = −(γ + 1)κγρ γ −2 < 0, ϕ1 (U , α ) < ρmax (U , α ) < ϕ2 (U , α ) it holds that g U , ϕ1 (U , α ) > g U , ρmax (U ) = > g U , ϕ1 (U , α ) The last two inequalities justify the statement in (b) Moreover, F (U , ρ0 ; α ) = ρ0 u (1 − α0 /α ) > iff a > α0 , which proves (3.15), and shows that ρ0 is located outside of the interval [ϕ1 (U , α ), ϕ2 (U , α )] in the opposite case Since γ −1 ∂ F (U , ρ0 ; α ) u 20 − κγρ0 = ∂ρ u0 < iff U ∈ G , which, together with the earlier observation, implies (3.16) We next check (3.17) for a = α0 It comes from the definition of √ κγ ρ ∗ γ +1 αmin (U ) that αmin (U ) < α0 if and only if > ρ0 |u |, that can be equivalently written as Q (m) := γ +1 γ −1 where m := ρ0 1−γ m − (κγ ) γ +1 m γ +1 + γ −1 > 0, κγ (γ + 1) /u 20 Then, we can see that Q (1/κγ ) = 0, (3.19) which, in particular shows that the second equation in (3.17) holds, since (ρ0 , u ) ∈ C± for m = 1/κγ Moreover, d Q (m) dm = γ +1 1−γ − (κγ m) γ +1 , which is positive for m > 1/κγ and negative for m < κγ This together with (3.19) establish the first statement in (3.17) The third statement in (3.17) is straightforward This completes the proof of Lemma 3.3 ✷ To select a unique stationary wave, we need the following so-called Monotonicity Criterion The relationships (3.6) also define a curve ρ → α = α (U , ρ ) So we require that Monotonicity Criterion The volume fraction α = α (U , ρ ) must vary monotonically between the two values where ρ1 is the ρ -value of the corresponding state of a stationary wave having U as one state ρ0 and ρ1 , A similar criterion was used by Kröner, LeFloch, and Thanh [25,21,22], Isaacson and Temple [16,17] Geometrically, we can choose either ϕ1 or ϕ2 in the domains G , G , G using the following lemma Lemma 3.4 The Monotonicity Criterion is equivalent to saying that any stationary shock does not cross the boundary C In other words: (i) If U ∈ G ∪ G , then only the zero ϕ (U , α ) = ϕ1 (U , α ) is selected (ii) If U ∈ G , then only the zeros ϕ (U , α ) = ϕ2 (U , α ) is selected M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 Proof The second equation of (3.6) determines the u-value as u = u (ρ ) Taking the derivative with respect to equation α u (ρ )ρ 711 ρ in the = (α0 u ρ0 )2 , we get α (ρ )α (ρ )(u ρ )2 + 2α (u ρ ) u (ρ ) ρ + u (ρ ) = (3.20) Thus, to prove the lemma, it is sufficient to show that the factor (u (ρ ) ρ + u (ρ )) remains of a constant sign whenever (ρ , u ) remains in the same domain Indeed, assume for simplicity that u > 0, then u (ρ )ρ + u (ρ ) = = −κγρ γ −1 u +u u − κγρ γ −1 u , which remains of a constant sign as long as (ρ , u ) remain in the same domain This completes the proof of Lemma 3.4 ✷ A well-balanced scheme based on stationary waves In this section we will propose a numerical method to approximate the solution of the BN model (1.1)–(1.2) Our method is relying on a given standard numerical scheme As seen earlier, the system is regrouped into three subsystems which have different behaviors For the first subsystem of the governing equations in the gas phase, the equilibrium states representing the effect of the source terms to the system will then be cooperated into the standard scheme The second subsystem has the conservative form and is treated in a usual way For the third subsystem of the compaction dynamics equation, we argue that the discretized equation in fact has more regularity than it is theoretically supposed due to the separation of waves and the properties of the solid contact Then, we invoke the Engquist–Osher scheme for this subsystem Let us now present the details Given a uniform time step t, and a spacial mesh size x, setting x j = j x, j ∈ Z , and tn = n t , n ∈ N, we denote U nj to be an approximation of the exact value U (x j , tn ) A CFL condition is also required on the mesh sizes: λ max |u g | + p g (ρ g ), |u s | + U p s (ρs ) < 1, λ := t x (4.1) To discretize the equations on the gas phase, or more precisely, the first subsystem (3.1), we use the following strategy which consists of two steps: (i) First, we deal with the impact of the change of the volume fraction If the volume fraction changes, the nonconservative term p g ∂x α g is absorbed into stationary waves that produce equilibrium states Thus, these equilibrium states are obtained as the result of volume fraction change (ii) Second, the equilibrium states obtained from the first step will move according to the governing equation where the volume fraction is constant This enables us to eliminate the volume fraction on both sides of the equations so that the subsystem becomes the usual isentropic gas dynamics Thus, we set the dependent conservative variable v= ρg ρg u g (4.2) We then take a suitable standard numerical flux for isentropic gas dynamic equations g ( v , w ) The first component of the well-balanced scheme for the first subsystem is defined by v nj +1 = v nj − λ g v nj , v nj+1,− − g v nj−1,+ , v nj (4.3) The states v nj,± , j ∈ Z , n ∈ N are defined as follows Set v nj+1 = ρ ng, j+1 ρ ng, j+1 ung , j+1 (4.4) We need to make the scheme always well-defined Therefore, it is necessary to define the approximate “relaxation” value for the volume fraction αng,,Relax = max αng, j , αmin αng, j +1 , ρ ng, j +1 , ung , j +1 , j (4.5) 712 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 where αmin is defined by (3.10) Here the “relaxation” means that the volume fraction sudden decreases We first compute the corresponding value α g should be prevented from large ρ ng, j+1,− := ϕi αng, j+1 , ρ ng, j+1 , ung , j+1 , αng,,Relax j (4.6) where ϕi , i = 1, 2, are defined as in Lemma 3.2 The choice of ϕ1 or Lemma 3.4) This can be performed in the following procedure ϕ2 in (4.6) has to satisfy the Monotone Criterion (see • if the point (ρ ng, j +1 , ung , j+1 ) belongs to either the lower region G or the upper region G in the (ρ , u )-plane defined n,Relax n n n ) is selected; ( g , j +1 , g , j +1 , u g , j +1 , g , j n n if the point ( g , j +1 , u g , j +1 ) belongs to the middle region G in the ( , u )-plane, then in (4.6) the value ( ng, j +1 , n , ung , j +1 , ng,,Relax ) is selected; g , j +1 j if the point ( ng, j +1 , ung , j +1 ) belongs to the boundary curve C , then the selection can be made such that the resulted point by the stationary contact is located in the same region as its nearest neighboring point ( ng, j +l , ung , j +l ), by (3.11), then in (4.6) the value • ρ • ϕ α ρ α ρ α ρ ρ ϕ α ρ l = 0, ±1, ±2, Second, we compute the value u = ung , j +1,− using (3.7): ung , j +1,− = αng, j+1 ρ ng, j+1 ung , j+1 αng,,Relax ρ ng, j+1,− j (4.7) (4.8) The state v nj+1,− is defined by ρ ng, j+1,− v nj+1,− = ρ ng, j+1,− ung , j+1,− Similarly, set v nj−1 = ρ ng, j−1 ρ ng, j−1 ung , j−1 (4.9) We define the approximate “relaxation” value for the volume fraction αng,,Relax = max αng, j , αmin αng, j −1 , ρ ng, j −1 , ung , j −1 j (4.10) We first compute the corresponding value ρ ng, j−1,+ := ϕi αng, j−1 , ρ ng, j−1 , ung , j−1 , αng,,Relax j satisfying the Monotone Criterion i = 1, Also, the value can be performed in the following procedure (4.11) ϕi is the one that fulfills the Monotone Criterion, i = 1, This • if the point (ρ ng, j −1 , ung , j−1 ) belongs to either the lower region G or the upper region G in the (ρ , u )-plane defined n,Relax n n n ) is selected; ( g , j −1 , g , j −1 , u g , j −1 , g , j n n if the point ( g , j −1 , u g , j −1 ) belongs to the middle region G in the ( , u )-plane, then in (4.11) the value ( ng, j −1 , n,Relax n n ) is selected; g , j −1 , u g , j −1 , g , j n if the point ( g , j −1 , ung , j −1 ) belongs to the boundary curve C , then the selection can be made such that the resulted point by the stationary contact is located in the same region as its nearest neighboring point ( ng, j −l , ung , j −l ), by (3.11), then in (4.11) the value • ρ • ϕ α ρ ρ α ρ α ρ ϕ α ρ l = 0, ±1, ±2, Then, the value u = ung , j −1,+ is computed using (3.7) as ung , j −1,+ = αng, j−1 ρ ng, j−1 ung , j−1 αng,,Relax ρ ng, j−1,+ j (4.12) (4.13) The state v nj−1,+ is defined by v nj−1,+ = ρ ng, j−1,+ ρ ng, j−1,+ ung , j−1,+ M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 713 We now turn to deal with the second subsystem (3.2) which has the conservative form In this subsystem, we set w= αs ρs α g ρ g u g + αs ρs u s f 2(w ) = , αs ρs u s , α g (ρ g u 2g + p g ) + αs (ρs u 2s + p s ) (4.14) and we simply take a convenient standard numerical flux for the subsystem (3.2) For example, we may take a scheme involving the unknown function and the flux function only such as the Lax–Friedrichs scheme, the Lax–Wendroff scheme, or Richtmyer’s scheme, etc Finally, to complete the discretization of the whole model, we employ the technique in the Engquist–Osher scheme for the compaction dynamics equation (1.2) We first write u = max{u , 0} + min{u , 0} = u + + u − , (4.15) and then we apply the Engquist–Osher scheme for the compaction dynamics equation (1.2), which reads n n αnj +1 = αnj − λ u +, αnj − αnj−1 + u −, αnj+1 − αnj j j (4.16) Remark On one hand, as seen earlier, among elementary waves, the volume fractions change only across the solid contacts associated with the characteristic speed λ5 = u s On the other hand, across a solid contact, the solid velocity is constant For solutions with separated waves, this ensures that the nonconservative term u s ∂x α g can be discretized using the upwind scheme In the case where the exact solution containing waves that have the same speed, the scheme generates approximate solutions with separated waves Thus, the discretization techniques of the Engquist–Osher scheme is suitable for the compaction dynamics equation Remark Let us be given a stationary contact In the gas phase, it holds that αng, j+1 ρ ng, j+1 ung , j+1 = αng, j ρ ng, j ung , j , (ung , j +1 )2 + h g ρ ng, j +1 = (ung , j )2 + h g ρ ng, j This implies that ρ ng, j+1,− = ρ ng, j , ung , j +1,− = ung , j , ρ ng, j−1,+ = ρ ng, j , ung , j −1,+ = ung , j , so that v nj+1,− = v nj , v nj−1,+ = v nj This yields v nj +1 = v nj (4.17) Furthermore, in the mixture it is derived from (4.14) that the left-hand and the right-hand states of the contact satisfy α g ρ g u 2g + p g + αs ρs u 2s + p s = (4.18) which yields, since u s = α g ρ g u 2g + p g + αs p s = The last equation enables us to compute the exact pressure of the stationary contact in the solid phase From (4.17) and (4.18) we conclude that the scheme (4.3)–(4.13) captures exactly stationary waves In particular, it is a well-balanced scheme in the sense of [18,19] (the steady state solution is captured numerically either exactly or with at least a second order accuracy) Numerical experiments In this section we will present several numerical tests in which we compare the approximate solution and the exact Riemann solution We need to precise some details To fix the equations of state γ pk (ρk ) = κk ρk k , k = g , s, (5.1) 714 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 we take κ g = 0.4, κs = 1, γ g = 1.4, γs = 1.6 The exact solution can be obtained by considering the Riemann problem for (1.1)–(1.2) with the initial data of the form (α g0 , p g0 , u g0 , p s0 , u s0 )(x) = (α g L , p g L , u g L , p sL , u sL ) if x < 0, (α g R , p g R , u g R , p sR , u sR ) if x > (5.2) For simplicity, our method is composed with the Lax–Friedrichs scheme This means that on the first subsystem for the gas phase, the scheme reads U nj +1 = U nj+1,− + U nj−1,+ − λ f U nj+1,− − f U nj−1,+ , (5.3) and on the second subsystem – Conservation Law of mass of the solid phase and Conservation Law of the momentum of the mixture – the scheme is standard: U nj +1 = U nj+1 + U nj−1 − λ f U nj+1 − f U nj−1 (5.4) The solution will be computed on the interval [−1, 1] of the x-space, where for Test we use 1000 mesh points, and for Tests 2, 3, we use different mesh-sizes (1000, 2000, and 3000 mesh points) for the space discretization, and at the time t = 0.1 To make it clear we also provide the description of the exact Riemann solution Observe that through each separate proper elementary wave (shock wave or rarefaction wave) associated with the characteristic fields of one phase, i.e the characteristic fields involve only the quantities of the phase, the pressure and the velocity in the other phase not change Notations: • R i (U − , U + ) stands for an i-rarefaction wave connecting the left-hand state U − to the right-hand state U + , i = 1, 2, 3, 4; S i (U − , U + ) denotes an i-shock wave from the left-hand state U − to the right-hand state U + , i = 1, 2, 3, 4; W (U − , U + ) denotes a solid contact from the left-hand state U − to the right-hand state U + ; • W i (U , U ) → W j (U , U ) denotes an i-wave from the left-hand state U to the right-hand state U , followed by a j-wave from the left-hand state U to the right-hand state U , i , j = 1, 2, 3, 4, 5, W = R , S As seen latter, numerical tests show that our method can give a good approximation of the solution of the BN model (1.1)–(1.2) 5.1 Numerical Test 1: A test for stationary contacts As argued earlier, our method is capable of maintaining equilibrium states We illustrate this claim by carrying out the following test for a stationary contact Consider the Riemann problem for (1.1)–(1.2) where the Riemann data given by (α g L , p g L , u g L , p sL , u sL ) = (0.5, 0.4, 15, 3.0314331, 0), (α g R , p g R , u g R , p sR , u sR ) = (0.6, 0.3096989, 15.006578, 3.7014024, 0) (5.5) It is not difficult to check that in this case the Riemann solution is a stationary contact The Figs 2, and show that the stationary contact is well captured 5.2 Numerical Test In this test, we will approximate a nonstationary solution The solution is made from discontinuities: shock waves and the solid contact Precisely, we consider the Riemann problem for (1.1)–(1.2) with the Riemann data (α g L , p g L , u g L , p sL , u sL ) = (0.2, 0.066328921, −3.9737449, 2.1212506, 1.3373023), (α g R , p g R , u g R , p sR , u sR ) = (0.21, 0.37301208, −4.0055298, 0.30456574, −1.1053303) Set U L = (0.2, 0.066328921, −3.9737449, 2.1212506, 1.3373023), U = (0.2, 0.17504307, −4.4166667, 2.1212506, 1.3373023), U = (0.2, 0.4, −4, 2.1212506, 1.3373023), U = (0.2, 0.4, −4, 3.0314331, 1), U = (0.21, 0.37301208, −4.0055298, 3.0649166, 1), U R = (0.21, 0.37301208, −4.0055298, 0.30456574, −1.1053303) (5.6) M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 Fig Test 1: Volume fraction of gas phase Fig Test 1: Pressures on both phases Fig Test 1: Velocities on both phases 715 716 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 Fig Test 2: Wave structure of the exact Riemann solution Fig Test 2: The exact Riemann solution Fig Test 2: Approximation of exact solution: volume fraction of the gas phase Fig Test 2: Approximation of exact solution: pressure of the gas phase The Riemann solution is a 1-shock wave from U L to U , followed by a 2-shock wave from U to U , followed by a 3-shock wave from U to U , followed by a solid contact from U to U and then followed by a 4-shock wave from U to U R : S (U L , U ) → S (U , U ) → S (U , U ) → W (U , U ) → S (U , U R ) The structure of the Riemann solution is shown in Fig and the solution is displayed in Fig Figs 7, 8, 9, 10, and 11 show that approximate solutions are closer to the exact solution when the mesh-size gets smaller 5.3 Numerical Test In this test, we will approximate a nonstationary solution Here, we consider the Riemann problem for (1.1)–(1.2) with the Riemann data (α g L , p g L , u g L , p sL , u sL ) = (0.8, 0.022033232, −2.0382659, 2.1212506, 1.3373023), (α g R , p g R , u g R , p sR , u sR ) = (0.82, 0.14617279, −2.0036438, 13.079606, 2.5384615) Set U L = (0.8, 0.022033232, −2.0382659, 2.1212506, 1.3373023), U = (0.8, 0.058146048, −2.4166667, 2.1212506, 1.3373023), (5.7) M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 717 Fig Test 2: Approximation of exact solution: velocity of the gas phase Fig 10 Test 2: Approximation of exact solution: pressure of the solid phase Fig 11 Test 2: Approximation of exact solution: velocity of the solid phase Fig 12 Test 3: Wave structure of the exact Riemann solution U = (0.8, 0.15157166, −2, 2.1212506, 1.3373023), U = (0.8, 0.15157166, −2, 3.0314331, 1), U = (0.82, 0.14617279, −2.0036438, 3.3517208, 1), U R = (0.82, 0.14617279, −2.0036438, 13.079606, 2.5384615) The Riemann solution is a 1-shock wave from U L to U , followed by a 2-rarefaction wave from U to U , followed by a 3-shock wave from U to U , followed by a solid contact from U to U and then followed by a 4-rarefaction wave from U to U R : S (U L , U ) → R (U , U ) → S (U , U ) → W (U , U ) → R (U , U R ) The Riemann solution has the structure as shown in Fig 12 and is displayed in Fig 13 Figs 14, 15, 16, 17, and 18 show that approximate solutions are closer to the exact solution when the mesh-size gets smaller 5.4 Numerical Test This test also deals with the approximation of a non-stationary solution We consider the Riemann problem for (1.1)–(1.2) with the Riemann data 718 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 Fig 13 Test 3: The exact Riemann solution Fig 14 Test 3: Approximation of exact solution: volume fraction of the gas phase Fig 15 Test 3: Approximation of exact solution: pressure of the gas phase Fig 16 Test 3: Approximation of exact solution: velocity of the gas phase Fig 17 Test 3: Approximation of exact solution: pressure of the solid phase Fig 18 Test 3: Approximation of exact solution: velocity of the solid phase M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 719 (α g L , p g L , u g L , p sL , u sL ) = (0.5, 0.15157166, 3, 2.1212506, −2.6626977), (α g R , p g R , u g R , p sR , u sR ) = (0.51, 0.048187263, 3.3624504, 12.2833, −1.4615385) (5.8) Set U L = (0.5, 0.15157166, 3, 2.1212506, −2.6626977), U = (0.5, 0.15157166, 3, 3.0314331, −3), U = (0.51, 0.14737871, 3.0014115, 3.0902489, −3), U = (0.51, 0.14737871, 3.0014115, 12.2833, −1.4615385), U = (0.51, 0.055846086, 3.421381, 12.2833, −1.4615385), U R = (0.51, 0.048187263, 3.3624504, 12.2833, −1.4615385) The Riemann solution is a 3-shock wave from U L to U , followed by a solid contact from U to U , followed by a 4rarefaction wave from U to U , followed by a 1-rarefaction wave from U to U and then followed by a 2-shock wave from U to U R : S (U L , U ) → W (U , U ) → R (U , U ) → R (U , U ) → S (U , U R ) The Riemann solution has the structure as shown in Fig 19 and is displayed in Fig 20 Fig 19 Test 4: Wave structure of the exact Riemann solution Fig 20 Test 4: The exact Riemann solution Fig 21 Test 4: Approximation of exact solution: volume fraction of the gas phase Fig 22 Test 4: Approximation of exact solution: pressure of the gas phase 720 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 Fig 23 Test 4: Approximation of exact solution: velocity of the gas phase Fig 24 Test 4: Approximation of exact solution: pressure of the solid phase Fig 25 Test 4: Approximation of exact solution: velocity of the solid phase Figs 21, 22, 23, 24, and 25 show that approximate solutions are closer to the exact solution when the mesh-size gets smaller Test show that the stationary contact was captured exactly Tests 2, 3, show that the approximate solutions converge to the exact solution This justifies the efficiency of our well-balanced numerical method Acknowledgements The authors would like to thank the referees for their very constructive comments References [1] A Ambroso, C Chalons, F Coquel, T Galié, Relaxation and numerical approximation of a two-fluid two-pressure diphasic model, Math Mod Numer Anal 43 (2009) 1063–1097 [2] N Andrianov, G Warnecke, The Riemann problem for the Baer–Nunziato model of two-phase flows, J Comput Phys 195 (2004) 434–464 [3] E Audusse, F Bouchut, M.-O Bristeau, R Klein, B Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J Sci Comput 25 (6) (2004) 2050–2065 [4] M.R Baer, J.W Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Int J MultiPhase Flow 12 (6) (1986) 861–889 [5] R Botchorishvili, O Pironneau, Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws, J Comput Phys 187 (2003) 391–427 [6] R Botchorishvili, B Perthame, A Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math Comput 72 (2003) 131–157 M.D Thanh et al / Applied Numerical Mathematics 61 (2011) 702–721 721 [7] F Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Frontiers in Mathematics Series, Birkhäuser, 2004 [8] J.B Bzil, R Menikoff, S.F Son, A.K Kapila, D.S Steward, Two-phase modelling of a deflagration-to-detonation transition in granular materials: a critical examination of modelling issues, Phys Fluids 11 (2) (1999) 378–402 [9] A Chinnayya, A.-Y LeRoux, N Seguin, A well-balanced numerical scheme for the approximation of the shallow water equations with topography: the resonance phenomenon, Int J Finite (4) (2004) 1–33 [10] G Dal Maso, P.G LeFloch, F Murat, Definition and weak stability of nonconservative products, J Math Pures Appl 74 (1995) 483–548 [11] D.A Drew, S.L Passman, Theory of Multicomponent Fluids, Springer-Verlag, New York, 1999 [12] T Gallouët, J.-M Hérard, N Seguin, Numerical modeling of two-phase flows using the two-fluid two-pressure approach, Math Models Methods Appl Sci 14 (5) (2004) 663–700 [13] P Goatin, P.G LeFloch, The Riemann problem for a class of resonant nonlinear systems of balance laws, Ann Inst H Poincaré Anal Non Lineaire 21 (2004) 881–902 [14] J.M Greenberg, A.Y LeRoux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J Numer Anal 33 (1996) 1–16 [15] J.M Greenberg, A.Y LeRoux, R Baraille, A Noussair, Analysis and approximation of conservation laws with source terms, SIAM J Numer Anal 34 (1997) 1980–2007 [16] E Isaacson, B Temple, Nonlinear resonance in systems of conservation laws, SIAM J Appl Math 52 (1992) 1260–1278 [17] E Isaacson, B Temple, Convergence of the × Godunov method for a general resonant nonlinear balance law, SIAM J Appl Math 55 (1995) 625–640 [18] S Jin, X Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J Comput Math 22 (2004) 230–249 [19] S Jin, X Wen, Two interface type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations, SIAM J Sci Comput 26 (2005) 2079–2101 [20] S Karni, G Hernández-Duenas, A hybrid algorithm for the Baer–Nunziato model using the Riemann invariants, J Sci Comput 45 (2010) 382–403 [21] D Kröner, M.D Thanh, Numerical solutions to compressible flows in a nozzle with variable cross-section, SIAM J Numer Anal 43 (2) (2005) 796–824 [22] D Kröner, P.G LeFloch, M.D Thanh, The minimum entropy principle for fluid flows in a nozzle with discontinuous crossection, Math Mod Numer Anal 42 (2008) 425–442 [23] M.-H Lallemand, R Saurel, Pressure relaxation procedures for multiphase compressible flows, INRIA report No 4038, 2000 [24] P.G LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Math and Its Appl., Minneapolis, preprint, 593, 1989 [25] P.G LeFloch, M.D Thanh, Properties of Rankine–Hugoniot curves for Van der Waals fluid flows, Jpn J Ind Appl Math 20 (2) (2003) 211–238 [26] P.G LeFloch, M.D Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Commun Math Sci (4) (2003) 763–797 [27] P.G LeFloch, M.D Thanh, The Riemann problem for shallow water equations with discontinuous topography, Commun Math Sci (4) (2007) 865–885 [28] S.T Munkejord, Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation, Comput Fluids 36 (2007) 1061–1080 [29] V.H Ransom, D.L Hicks, Hyperbolic two-pressure models for two-phase flow, J Comput Phys 53 (1) (1984) 124–151 [30] R Saurel, R Abgrall, A multi-phase Godunov method for compressible multifluid and multiphase flows, J Comput Phys 150 (1999) 425–467 [31] R Saurel, R Abgrall, A simple method for compressible multi-fluid flows, SIAM J Sci Comput 21 (1999) 1115–1145 [32] D.W Schwendeman, C.W Wahle, A.K Kapila, The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow, J Comput Phys 212 (2006) 490–526 [33] B.H Stewart, B Wendroff, Two-phase flow: models and methods, J Comput Phys 56 (1984) 363–409 [34] M.D Thanh, The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J Appl Math 69 (6) (2009) 1501–1519 [35] M.D Thanh, A Izani Md Ismail, Well-balanced scheme for a one-pressure model of two-phase flows, Phys Scr 79 (2009) 065401, 7pp [36] M.D Thanh, Md Fazlul Karim, A Izani Md Ismail, Well-balanced scheme for shallow water equations with arbitrary topography, Int J Dyn Syst Differ Equ (2008) 196–204 [37] X Wen, A steady state capturing and preserving method for computing hyperbolic systems with geometrical source terms having concentrations, J Comput Phys 219 (2006) 322–390  ... product of a discontinuous function and the (partial) derivative of another quantity that might be discontinuous as well Mathematical formulation of nonconservative systems of balance laws was introduced... discretization of nonconservative terms therefore has been an attractive topic for many years Basically, a good numerical method for a nonconservative system should give a good approximation of the exact... Relaxation and numerical approximation of a two-fluid two-pressure diphasic model, Math Mod Numer Anal 43 (2009) 1063–1097 [2] N Andrianov, G Warnecke, The Riemann problem for the Baer–Nunziato model