Đang tải... (xem toàn văn)
DSpace at VNU: Numerical treatment of nonconservative terms in resonant regime for fluid flows in a nozzle with variable...
Computers & Fluids 66 (2012) 130–139 Contents lists available at SciVerse ScienceDirect Computers & Fluids j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p fl u i d Numerical treatment of nonconservative terms in resonant regime for fluid flows in a nozzle with variable cross-section Mai Duc Thanh a,⇑, Dietmar Kröner b a b 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 a r t i c l e i n f o a b s t r a c t Article history: Received October 2011 Received in revised form June 2012 Accepted 19 June 2012 Available online July 2012 When data are on both sides of the resonant surface, existing numerical schemes often give unsatisfactory results This phenomenon is probably caused by the truncation errors, which are added up to states near the resonant surface that could shift the approximate states into a wrong side of the resonant surface In this paper, we enhance the well-balanced scheme constructed in an earlier work with a computing corrector in the computing algorithm that selects the admissible equilibrium state We build up two computing correctors of different types: one depends on the mesh-size and the other depends on the time iteration number Each of these correctors will help the algorithm select the correct equilibrium state when there are two possible states Moreover, we also improve the computational method solving the nonlinear equation that determines the equilibrium states by driving an equivalent form of the equation such that the Newton–Raphson method can work perfectly Numerical tests show that our well-balanced scheme equipped with each of the above two computing correctors gives good approximations for initial data in resonant regime Ó 2012 Elsevier Ltd All rights reserved Keywords: Numerical treatment Well-balanced scheme Fluid dynamics Nozzle Hyperbolic conservation law Source term Shock wave Stationary wave Introduction We are interested in the numerical treatment of the nonconservative term of the following model of fluid flows in a nozzle with variable cross-section @ t aqị ỵ @ x aquị ẳ 0; @ t aquị ỵ @ x aqu2 ỵ pịị ẳ p@ x a; @ t aqeị ỵ @ x auqe ỵ pịị ẳ 0; x R; ð1:1Þ t > 0; where a ¼ aðxÞ; x R represents the cross-section, q is the density, u is the velocity, e is the internal energy, T is the pressure, S is the entropy, and e ẳ e ỵ u2 =2 is the total energy A standard way to put the system (1.1) under the framework of hyperbolic conservation laws is to supplement it with an additional trivial equation @ t a ẳ 0; 1:2ị see [26,27] In the literature, numerical treatments of nonconservative systems such as (1.1) have attracted lots of attentions of scientists Most existing schemes could succeed to approximate the exact solutions in strictly hyperbolic domains In particular, in [24] we build a well-balanced numerical scheme that can capture equilib⇑ Corresponding author E-mail addresses: mdthanh@hcmiu.edu.vn (M.D Thanh), dietmar@mathematik uni-freiburg.de (D Kröner) 0045-7930/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compfluid.2012.06.021 rium states and provides us with good approximations for data in strictly hyperbolic domains In that work, the nonconservative term is made absorbed by admissible stationary contacts that result equilibrium states See the references therein for related works However, when data are on both sides of the resonant surface at which the system fails to be strictly hyperbolic, numerical oscillations and divergence could be observed For example, when a rarefaction wave in one side is attached to a stationary contact that jumps to the other side By investigating the selection procedure which chooses the admissible equilibrium point resulted by a stationary wave at the resonance surface, we discover that a computing selection procedure may be different from the theoretical procedure, probably due to the propagation of errors More precisely, errors adding to a state belonging to one side of the resonant surface may result an approximate state that falls into the other side of the resonant surface As well-known, the most complicated situation of the system (1.1) occurs across the resonant surface, where the Riemann problem may admit one, two or three solutions of different structures, see [38] Consequently, the well-balanced scheme would unable to produce a good approximation the exact solution when errors propagate across the resonant surface To deal with the above problem, we enforce our well-balanced scheme in [24] with a computing corrector that enables the algorithm computing the admissible state across the resonant surface to select the right state We will present in this paper two computing correctors of different types: one corrector depends on the mesh-size and the other one depends on the time 131 M.D Thanh, D Kröner / Computers & Fluids 66 (2012) 130–139 iteration number This work is motivated from the analysis of a situation of a Riemann solution where a rarefaction wave in one side of the resonant surface reaches the resonant surface and is then followed up by a stationary contact that jumps to the other side of the resonant surface This is the most challenging situation since small errors may result huge impact for numerical approximation In all other situations, even when data are on both sides of the resonant surface, small errors not play any significant role in our well-balanced scheme [24] This is because the approximate state stays in the same side of the exact state and the selection procedure for the admissible wave works Therefore, our well-balanced scheme in [24] works properly Besides, we also develop in this work a robust numerical method to compute admissible stationary contacts The nonlinear equation for the density of the admissible stationary contact will then be transformed into a convex form so that the Newton–Raphson method works Furthermore, we also describe an computing algorithm for selecting the admissible stationary contacts Numerical tests show that our well-balanced method after cooperating one of the above-mentioned two computing correctors provides us with good approximations of the exact solutions of (1.1) for data on both side of the resonance surface Moreover, in the recent interesting work [33], by presenting a systematic comparison of admissible configurations between the one-dimensional nonconservative model and the axisymmetric conservative Euler system, the authors conclude that there is a very good correspondence between the two models when the solutions of the axisymmetric model possesses straight longitudinal shocks, so that no noticeable transversal shock perturbs the solution Therefore, we also include several tests where the exact solutions were considered through the comparison in [33] There have been many works concerning the model (1.1) in the literature First, the model (1.1) can theoretically be understood in the sense of nonconservative products, see [11] The analysis of shock waves and other waves of (1.1), and related models can be seen in [27,31,28,38,20,19,15,2,3,29] Numerical approximations for the model of fluid flows in a nozzle with variable cross-section were studied in [24,23,33,21,22] Well-balanced schemes for shallow water equations was considered by an early work [17], and then developed in [8,39,21,22,14,34,30] Well-balanced numerical schemes for a single conservation law with source term were studied in [18,6,7,16,4] Well-balanced schemes for multi-phase flows and other models were studied in [5,25,36,1,40–42] Numerical schemes for nonconservative hyperbolic systems were considered in [32,37,9,35,12,13,10] See also the references therein The organization of this paper is as follows In Section we provides basic properties of the model (1.1) Section is devoted to equilibrium states, where characterization of roots of the nonlinear equations determining equilibrium states are summarized, and the computing algorithm for the admissible root is given In Section we review our well-balanced scheme and introduce two computing correctors Section is devoted to numerical tests Finally in Section we will draw conclusions on our results and we also address some future related study Take the variable U ẳ q; u; S; aị The system (1.1), (1.2) can be written in the vector form U t ỵ AUịU x ẳ 0; where u B pq Bq AUị ẳ B B @0 q u pS u 0 k0 ¼ 0; where c > 1; C v > and Sà are constants Then pq q; Sị ẳ cpq; Sị pq; Sị ; pS q; Sị ẳ : Cv q C 0C C: C 0A k1 ẳ u c; 2:2ị k2 ẳ u; k3 ẳ u ỵ c; 2:3ị where c is the local sound speed c¼ pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pq ¼ cpðq; SÞ=q: The corresponding eigenvectors of AðUÞ can be chosen as B B r0 ¼ B B @ u2 q Àupq aðpq À u2 Þ C C C; C A B Àc C C B r1 ¼ B C; @ A q 1 ÀpS B C C B r2 ¼ B C; @ pq A 0 q BcC B C r ¼ B C: @0A 0 Since the characteristic field associated with k0 may coincide with any other field, the system (2.1) is not strictly hyperbolic Set G1 ẳ fU : k0 Uị < k1 Uị < k2 Uị < k3 Uịg; G2 ẳ fU : k1 ðUÞ < k0 ðUÞ < k2 ðUÞ < k3 ðUÞg; G3 ẳ fU : k1 Uị < k2 Uị < k0 Uị < k3 Uịg; G4 ẳ fU : k1 ðUÞ < k2 ðUÞ < k3 ðUÞ < k0 ðUÞg: 2:4ị Rỵ ẳ fU : k1 Uị ẳ k0 Uịg; R0 ẳ fU : k2 Uị ẳ k0 Uịg; R ¼ fU : k3 ðUÞ ¼ k0 ðUÞg; R ¼ Rỵ [ R [ R0 : In what follows we will refer to R as the resonant surface Equilibrium states Let us be given a state U ¼ ðq0 ; u0 ; a0 Þ with a level of cross-section a0 and another level cross-section a1 As in [24], a state U ¼ ðq1 ; u1 ; a1 Þ with the level cross-section a1 which can be connected with U via a stationary wave is determined by the system ẵaqu ẳ 0; ! u2 ỵ hq; S0 ị ẳ 0; 3:1ị ẵS ẳ 0; where h is the specific enthalpy (3.16), which is given as a function h ẳ hq; Sị by hq; Sị ẳ c exp S À Sà c p ¼ pq; Sị ẳ c 1ị exp q; Cv q uq a The matrix AðUÞ in (2.2) admits four real eigenvalues, Preliminaries Let us consider a polytropic fluid where the equation of state is given in the form ð2:1Þ S À Sà cÀ1 q : Cv ð3:2Þ Set ASị ẳ c 1ị exp S S ; Cv j ẳ AS0 ị; l ẳ 2jc : cÀ1 ð3:3Þ In [24], we solve for q from the nonlinear equation lqcỵ1 u20 ỵ lq0c1 q2 ỵ a u q 2 0 ¼ 0: a ð3:4Þ 132 M.D Thanh, D Krưner / Computers & Fluids 66 (2012) 130–139 q0 < u1 ðU ; aÞ for U G1 [ G4 ; q0 > u2 ðU ; aÞ for U G2 [ G3 : The function on the left-hand side of (3.4), unfortunately, is not convex Therefore, the Newton–Raphson method may not work We look for an equivalent form of (3.4) into a convex form 3.1 Characterization of the roots To characterize the roots of the nonlinear Eq (3.4), we will rewrite (3.4) in a form that is convenient for investigating properties of these roots Employing the techniques in [28], we transform (3.4) into the following equivalent form UU ; a; qị :ẳ sgnu0 ị u20 À lðqcÀ1 À qc0À1 Þ 1=2 qÀ a0 u0 q0 ¼0 a ð3:5Þ As we will see later on, we can easily investigate properties of the function on the left-hand side of (3.5) The function q # UðU ; a; qÞ is defined for ð3:11Þ To select a unique physical root among the two possible roots, we need the following criterion ADMISSIBILITY CRITERION Along the stationary curve between left- and right-hand states of any stationary wave, the component a expressed as a function of q has to be monotone in q As shown in [38], the above Admissibility Criterion is equivalent to the condition that any stationary wave has to remain in the closure of a strictly hyperbolic domain Therefore, for U G1 [ G4 , we choose u1 ðU ; aÞ, and for U G2 [ G3 , we take u2 ðU ; aÞ, where U plays the role of a left-hand side state of the stationary contact 3.2 Computing algorithm @ UðU ; a; qÞ u20 À lðqcÀ1 À q0 ị jcqc1 ẳ 1=2 : @q u20 À lðqcÀ1 À qc0À1 Þ In this subsection we will describe the method to compute the admissible root among the two roots ui U ; aị; i ẳ 1; defined by (3.8) for given U and a The function q # UðU ; a; qÞ in (3.5) is not convex So we look for an equivalent form of (3.4) such that the resulted equation can be treated numerically by a standard favorite method such as the Newton–Raphson method Multiplying both sides of (3.4) by 1=q, we obtain Assume, for simplicity, that u0 > The last expression means that a u q 2 0 FU ; a; qị :ẳ lqc u20 ỵ lqc01 q ỵ ẳ 0; a q U ị :ẳ 06q6q l u20 c1 ỵ q0 c1 : A straightforward calculation shows that cÀ1 @ UðU ; a; qÞ > 0; @q @ UðU ; a; qÞ < 0; @q ð3:12Þ q < qmax ðU Þ; q > qmax U ị; where qmax U ị :ẳ c1 c1 : u20 ỵ lq0 lc ỵ 1Þ ð3:6Þ The function q # UðU ; a; qÞ 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 P amin U ị :ẳ a0 q0 ju0 j : p cỵ1 jcqmax U ị 3:7ị For u0 < 0, similar properties hold Thus, given U , a stationary shock issuing from U and connecting to some state U ẳ q; u; aị exists if and only if a P amin ðU Þ When a > amin ðU Þ, then there are exactly two values u1 ðU ; aÞ < qmax ðU Þ < u2 ðU ; aÞ such that UðU ; a; u1 U ; aịị ẳ UU ; a; u2 U ; aịị ẳ 0: ð3:8Þ As in [28], we obtain: qmax ðU Þ > q0 ; ðU Þ G1 [ G4 ; qmax ðU Þ < q0 ; ðU Þ G2 [ G3 ; qmax ðU Þ ẳ q0 ; U ị Cặ : a u q 2 dFðU ; a; qị 0 ẳ lcqc1 u20 ỵ lqc01 À ; dq a q2 a u q 2 d FðU ; a; qÞ 0 ẳ lcc 1ịqc2 ỵ > 0; q > 0: dq a q3 ð3:13Þ The function q # FðU ; a; qÞ attains a unique strictly minimum value at a point where its derivative given by (3.13) vanishes However, the analytic form of this minimum point is not available This raises a difficulty when applying the Newton–Raphson method, since we would not know which roots the method gives if we start the method at an arbitrary point In the following we will deal with the starting point of the Newton–Raphson method such that it will give us the admissible root First, it is not difficult to check that FðU ; a; qÞUðU ; a; qÞ < 0; (i) If a > a0 , then (ii) If a < a0 , then where l is defined by (3.3) Since Eq (3.12) is an equivalent form of Eq (3.5), it has the same roots under the same conditions as seen in the above argument The interesting is that the function on the lefthand side of (3.12) is strictly convex This enable us to apply the Newton–Raphson method to calculate its two roots u1 ðU ; aÞ and u2 ðU ; aÞ Indeed, a simple calculation gives ð3:9Þ The state ðu1 ðU ; aÞ; a0 u0 q0 =ðau1 ðU ; aÞÞÞ from the other side of a stationary jump from U belongs to G1 if u0 > 0, and belongs to G4 if u0 < 0, while the state ðu2 ðU ; aÞ; a0 u0 q0 =ðau2 ðU ; aÞÞÞ belongs to G2 if u0 > and belongs to G3 if u0 < In addition, it holds that u1 ðU ; aÞ < q0 < u2 ðU ; aÞ: q > 0; ð3:10Þ < q – ui U ; aị; i ẳ 1; 2: 3:14ị As observed in the previous subsection, we have two roots with FðU ; a; q0 Þ < 0; u1 ðU ; aÞ < q0 < u2 ðU ; aÞ: To get the root u1 ðU ; aÞ, we can use the Newton–Raphson method applied to the function q # FðU ; a; qÞ with a starting point less than u1 ðU ; aÞ How to choose such a point? Consider ’’small’’ values of q It follows from (3.14) that a u q 2 0 FðU ; a; qÞ > u20 ỵ lqc01 q ỵ P0 a q for 133 M.D Thanh, D Kröner / Computers & Fluids 66 (2012) 130–139 a0 u0 q0 ffi < u1 ðU ; aị: q q1 :ẳ q a u20 þ lq0cÀ1 ð3:15Þ Since FðU ; a; qÞ > 0; d FðU ; a; qÞ=dq2 > 0; for < q < u1 ðU ; aÞ and q1 < u1 ðU ; aÞ, the Newton–Raphson method starting at q1 will generate a monotone increasing sequence that converges to u1 ðU ; aÞ In a similar manner, to get the root u2 ðU ; aÞ, we can use the Newton–Raphson method with a starting point larger than u2 ðU ; aÞ To choose such a starting point, consider ‘‘large’’ values of q It follows from (3.14) that FðU ; a; qị > lqc u20 ỵ lq0c1 q P u0 l ỵ qc01 1=c1ị c ỵ 1=c1ị ẳ qmax U ; aị > u2 ðU ; aÞ: ð3:16Þ Since FðU ; a; qÞ > 0; 4.1 Well-balanced numerical scheme Given a time step Dt > and a spacial mesh size Dx Set xj ¼ jDx; j Z; t n ¼ nDt; n N; k¼ Dt ; Dx d FðU ; a; qÞ=dq2 > 0; for q > u2 ðU ; aÞ and q2 > u2 ðU ; aÞ, the Newton–Raphson method starting at q2 will generate a monotone decreasing sequence that converges to u2 ðU ; aÞ We can summarize the above argument in the following algorithm which describes the choice for a starting point in the Newton–Raphson method, applying to calculate the admissible root of the nonlinear Eq (3.12) 3.2.1 Algorithm of selecting admissible root A pseudo-code for Newton–Raphson method selecting the admissible root of Eq (3.12) can be described as follows We consider only for u0 > 0, since the case u0 can be treated similarly Algorithm q ẳ SelectingRootU ; aị while jFU ; a; qÞj < 1e À 12 À Á2 cÀ1 À a0 ua0 q0 q12 ; ¼ lcqcÀ1 À u20 ỵ lq0 dFU ;a;qị dq FU ;a;qị q ẳ q dFU ;a;qị=dq c1 FU ; a; qị ẳ lqc u20 ỵ lq0 q ỵ a0 ua0 q0 q1 ; end 4:2ị In the scheme (4.2), the states U njỵ1; ẳ q; qu; qeịnjỵ1; ; U nj1;ỵ ẳ q; qu; qeịnj1;ỵ are dened as follows First, observe that the entropy is constant across each stationary jump, we compute qnjỵ1; ; unjỵ1; from the equations anjỵ1 qnjỵ1 unjỵ1 ẳ anj qnjỵ1; unjỵ1; ; ẳ unjỵ1; ị2 unjỵ1 ị2 and we compute q anj1 qnj1 unj1 ẳ anj qnj1;ỵ unj1;ỵ ; ẳ unj1;ỵ ị2 ỵ hqnjỵ1 ị ỵ hqnjỵ1; ị; n n j1;ỵ ; uj1;ỵ 4:3ị from the equations unj1 ị2 ỵ hqnj1 ị ỵ hqnj1;ỵ ị: 4:4ị It was shown in our earlier work [24] that our scheme ((4.1)– (4.4)) is well-balanced For example, we can take the LaxFriedrichs numerical ux: gU; Vị ẳ If k1 U Þ P - Start at q1 : q ¼ q1 else - Starting at q2 : q ¼ q2 end À Á2 cÀ1 FðU ; a; qị ẳ lqc u20 ỵ lq0 q ỵ a0 ua0 q0 q1 ; 4:1ị Let g ẳ gU; VÞ be an underlying numerical flux of the usual gas dynamics equations, which corresponds to the case a constant in (1.1) Our well-balanced scheme for (1.1) is defined by U nỵ1 ẳ U nj k gU nj ; U njỵ1; ị gU nj1;ỵ ; U nj ị ; j for q P q2 :ẳ tionary wave that jumps into the other side of the resonant surface This is because error propagation probably cause the computing algorithm to select the wrong state in the wrong side To deal with this, in this section we will introduce two different computing correctors that enable the scheme to work properly in that case 1 f Uị ỵ f ðVÞÞ À ðV À UÞ: 2k ð4:5Þ 4.2 Computing correctors As indicated above, the selection of the states U nj;Ỉ involve the Monotone Criterion So, when a rarefaction wave started at U L G2 approaches and reaches the resonant surface Rỵ at a state U , it can be followed up by a stationary contact that is attached to it at U and jumps into U 2 G1 The attaching condition means that the characteristic speed k1 ðU Þ at the end of the rarefaction fan coincides with the discontinuity speed k0 ðU ; U ị ẳ Let U nj be an approximation of such an above mentioned state U of a rarefaction wave from U L G2 to U Rỵ We need to control the distance between U nj and the resonant surface Rỵ using k1 ðU nj Þ More precisely, whenever the following condition holds n k1 ðU nj Þ < Àdj ; Numerical schemes and computing correctors In [24], we built a numerical scheme for the model (1.1) Tests show that this scheme captures exactly equilibrium states and it provides us with convergence in strictly hyperbolic regions Moreover, it preserves the positivity of the density and possesses the numerical minimum entropy principle, see [23] As most existing schemes, its original version may fail to approximate solutions when data belong to both sides of the resonant surface In particular, the scheme may not work when a rarefaction wave is attached by a sta- n dj where > represents a computing corrector, we will take the root u1 U nj ; ajỵ1 ị to jump to U njỵ1; G1 Note that in this case U nj may belong to G1 or G2 , or Rỵ In the following we suggest two computing correctors: (I) A mesh-size dependent corrector: n dj ẳ Dxmax jki U nj ịj jqnjỵ1 qnj j ỵ junjỵ1 unj j ỵ jpnjỵ1 pnj j ; iẳ1;2;3 4:6ị 134 M.D Thanh, D Kröner / Computers & Fluids 66 (2012) 130–139 (II) Corrector depends on the number of the iterations: n dj ¼ maxiẳ1;2;3 jki U nj ịj p k n jỵ1 jq n jj q ỵ junjỵ1 unj j ỵ jpnjỵ1 together with the corresponding right-eigenvectors: pnj j ; ð4:7Þ where k is the number of iterations 1 q ffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi p B C B C B C r :¼ @ Àp0 qị A r :ẳ @ p0 qị A r :ẳ @ p0 qị A: ap0 qị 0 au À u qu q Set The way to take into account one of the above computing correctors can be described in the following algorithm pffiffiffiffiffiffi cÀ1 CỈ : u ẳ ặ jc q : We can see that 4.2.1 Algorithm of computing admissible root A pseudo-code for Newton–Raphson method selecting the admissible root of the Eq (3.12) can be described as follows We consider only for u0 > 0, since the case u0 can be treated similarly k1 ẳ k0 on Cỵ ; k2 ¼ k0 on CÀ : We consider the Riemann problem for (5.1), (1.2) with the Riemann data Algorithm q ¼ ComputingCorrectorðU ; a; dÞ & U ðxÞ ¼ If k1 ðU Þ P Àd – Start at q1 : q ¼ q1 else – Starting at q2 : q ¼ q2 end À Á2 c1 FU ; a; qị ẳ lqc u20 þ lq0 q þ a0 ua0 q0 q1 ; ð5:2Þ U ẳ 1:3783; 1:2616; 1ị while jFU ; a; qÞj < 1e À 12 À cÀ1 a u q ẳ lcqc1 u20 ỵ lq0 À a0 q12 ; U ¼ ð0:97819; 1:6161; 1:1ị; U ẳ 1:2304; 1:3387; 1:1ị: FU ;a;qÞ dFðU ;a;qÞ=dq À Á2 cÀ1 FðU ; a; qị ẳ lqc u20 ỵ lq0 q ỵ a0 ua0 q0 q1 ; 5:3ị The exact solution is a rarefaction wave from U L to U Cỵ , followed by a stationary wave from U to U , followed by a 1-shock from U to U , and then arrives at U R by a 3-shock Without a corrector, the well-balanced method with underlying Lax-Friedrichs scheme does not give a good approximation to the exact solution, see Fig end Test cases 5.2 Test 5.1 Test For Tests 1–3 below, the solution is evaluated for x ½À1; 1 with the mesh sizes of 1000 points and 3000 points, and at the time t ¼ 0:2 We take C:F:L ¼ 0:5: The following test is devoted to an isentropic ideal gas, where the pressure is given by p ¼ jqc ; x < 0; where G1 is the domain where k1 ðUÞ > 0, and G2 is the domain where k1 ðUÞ < and k2 ðUÞ > In this test, the Riemann data are taken on the opposite sides of the resonance curve in the q; uị-plane Set dFU ;a;qị dq qẳ q U L ẳ qL ; uL ; aL ị ẳ 3; 0:2; 1ị G2 ; U R ẳ qR ; uR ; aR ị ẳ 1:4; 1:5; 1:1ị G1 ; x > 0; In this test, we use the same data as in Test 1, and we equip our well-balanced scheme by the modified version of computing corrector I in (4.6) by neglecting the term of the pressure The underlying Lax-Friedrichs scheme is chosen This tests shows that our method provides us with a good approximations to the exact solution with 1000 and 3000 mesh points for the interval ½À1; 1, see Fig c > 1; j > and in the sequel, for simplicity we take j ¼ The governing equations of the model of the isentropic fluid in a nozzle with variable cross-section are given by @ t aqị ỵ @ x aquị ẳ 0; @ t aquị ỵ @ x aqu2 ỵ pịị ẳ p@ x a; x R; t > 0; ð5:1Þ Let us recall some basic properties of the model (5.1) The reader is referred to [28] for more details Taking the variable U ẳ q; u; aị, we can re-write the system (5.1),(1.2) in the form @ t U ỵ AUị @ x U ẳ 0; where u q qu=a @ AUị ẳ h0qị u A; 0 hqị ẳ jc c1 q : c1 The matrix AUị admits the following three eigenvalues k0 :ẳ 0; k1 :ẳ u p p0qị; k2 :ẳ u ỵ pffiffiffiffiffiffiffiffiffiffiffi p0ðqÞ; Fig Test Without the corrector, the well-balanced method with underlying Lax-Friedrichs scheme does not give a good approximation to the exact solution 135 M.D Thanh, D Kröner / Computers & Fluids 66 (2012) 130–139 Fig Test The well-balanced method with underlying Lax-Friedrichs scheme equipped by the corrector I in (4.6) gives a good approximation to the exact velocity R1 ðU L ; U Þ ! W ðU ; U Þ ! S1 ðU ; U Þ ! W ðU ; U Þ ! S3 ðU ; U R Þ; ð5:4Þ where R1 ðU L ; U Þ stands for a 1-rarefaction wave from U L to U ; W ðU ; U Þ stands for a stationary contact from U to U ; S1 ðU ; U Þ stands for a 1-shock from U to U ; W ðU ; U Þ stands for a 2-contact from U to U , and S3 ðU ; U R Þ stands for a 3-shock from U to U R The computing strategy of the states U i ; i ¼ 1; 2; 3; can be shown bellow Setting mSị ẳ c1=2 c 1ị1=2c exp S S ; 2cC v nSị ẳ 2mSị ; c1 wẳ c1 ; 2c 5:5ị Fig Test The configuration of the exact Riemann solution in the ðx; tÞ-plane 5.3 Test This test is devoted to a nonisentropic fluid, where we take c ¼ 1:4; C v ¼ 1; Sà ¼ 1: First, let us describe a way of computing exact solutions where data belong to both sides of the resonant surface and that a rarefaction wave in one side is attached by a stationary wave that jumps into the other side we can rewrite the resonant surface Rỵ as Rỵ : u ẳ mSịpw ; 5:6ị and the 1-wave rarefaction curve R1 ðU Þ as (see [38]) R1 U ị : u ẳ u0 nðS0 Þðpw À pw Þ; p p0 : ð5:7Þ The state U satisfies the equation w w u1 ẳ mSL ịpw ẳ uL nSL ịp1 À pL Þ ð5:8Þ which gives 5.4 Computing the typical Riemann solutions As indicated in [38], we can construct Riemann solutions by projecting all the wave curves in the ðp; uÞ-plan In particular, a Riemann solution can begins with a 1-rarefaction wave from U L ¼ ðpL ; uL ; SL ; aL Þ G2 and lasts until the rarefaction touches the resonant surface Rỵ at a state U ẳ p1 ; u1 ; SL ; aL ị since the entropy is a Riemann invariant At U , the characteristic speed k1 U ị ẳ and the solution can use a stationary contact to jump to a state U ¼ ðp2 ; u2 ; SL ; aR Þ G1 The intersection in the ðp; uÞ-plan of the forward 1-wave curve W ðU Þ and the backward 3-wave curve W ðU R Þ consists of one state U The Riemann solution is thus continued by a 1-wave from U to U This wave is a 1-shock if p2 < p3 and a 1-rarefaction wave otherwise In the computing strategy below we will choose a shock This 1-wave is followed by a 2-contact from U to U ¼ ðp3 ; u3 ; S4 ; aR Þ The solution then arrives at U R by a 3-wave from U This wave is a 3-shock if p3 > pR and a 3-rarefaction wave otherwise In the sequel we will choose a shock Thus, the solution has the form Fig p1 ¼ 1=w uL ỵ nSL ịpw L : mSL ị ỵ nSL Þ ð5:9Þ So, the state U ¼ ðp1 ; u1 ; SL ; aL Þ is determined by (5.9) and then (5.8) Next, to evaluate U , we rewrite the state U into the form U ¼ ðq1 ; u1 ; SL ; aL Þ with the q-component instead of the p-component using the equation of state q ẳ qp; Sị ẳ 1=c p Sà À S : exp Cv cÀ1 The state U ẳ q2 ; u2 ; SL ; aR ị is obtained by a stationary contact from U As indicated earlier, we calculate the q-component of U using the Newton–Raphson method The u-component of U is followed immediately Calculations for the Riemann solution continue with the determination of U as follows We rewrite U in the form U ¼ ðp2 ; u2 ; SL ; aR Þ In the ðp; uÞ-plane, the intersection of S ðU Þ and S ðU R Þ determines U It is necessary that uR < u2 As shown in [38], S ðU Þ and S ðU Þ are given by 136 M.D Thanh, D Kröner / Computers & Fluids 66 (2012) 130–139 Table States that separate the elementary waves of the exact Riemann solution in Test 3, see Fig UL U1 U2 U3 U4 UR q u p a 2.7766 1.6697 2.0779 1.8047 0.5 1.3306 1.8438 1.5738 1.5738 0.8 3.5111 1.7227 2.3427 2.3427 1 1.2 1.2 1.2 1.2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 À lịv S U ị : u ẳ u0 À ðp À p0 Þ ; p P p0 ; p ỵ lp0 s lịv ; p P p0 ; S ðU Þ : u ẳ u0 ỵ p p0 ị p ỵ lp0 lẳ c1 : cỵ1 It is easy to see that the function f1 ðpÞ in (5.12) is strictly increasing and strictly concave So, the root p ¼ p3 of the nonlinear Eq (5.12) can also be calculated using the Newton–Raphson method, where f10 ðpÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 À lÞv R p ỵ lpR ị1=2 ẳ p lịv R ỵ ỵ lịpR p ỵ lpR ị3=2 p lịv p ỵ lp2 ị1=2 ỵ p lịv ỵ ỵ lịp2 p ỵ lp2 ị3=2 > 0: Let us now consider the Riemann problem for (1.1),(1.2) with the Riemann data U xị ẳ & 5:10ị Thus, the state U satisfies the equations sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 À lÞv u ẳ u2 p p2 ị p ỵ lp2 s lịv R ; ẳ uR ỵ p pR ị p ỵ lpR p P maxfp2 ; pR g; 5:11ị where v ẳ 1=q It is derived from (5.11) that the p-component of U is the root of the nonlinear equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 À lÞv R lịv ỵ p p2 ị ỵ uR u2 ẳ 0: f1 pị ẳ p pR ị p ỵ lpR p ỵ lp2 5:12ị U L ¼ ðqL ; uL ; pL ; aL Þ ¼ ð5; 0:5; 8; 1Þ G2 ; x < 0; U R ¼ ðqR ; uR ; pR ; aR ị ẳ 1; 0:8; 1; 1:2ị G1 ; x > 0: ð5:13Þ In this test, the Riemann data are taken on the opposite sides of the resonance surface, where U L G2 and U R G1 The solution has the form (5.4), where the states that separate elementary waves of the Riemann solution are given in Table Our well-balanced scheme ((4.1)–(4.5)) equipped by either Corrector I in (4.6) or Corrector II in (4.7) gives good approximations as indicated in Fig 5.5 Test In this test, we consider a very interesting case where the exact Riemann solution may contain three waves of the same zero speed The states that determine the elementary waves of the exact Fig Test Our well-balanced method equipped by a computing corrector gives a good approximation to the exact pressure 137 M.D Thanh, D Kröner / Computers & Fluids 66 (2012) 130–139 Table States that separate the elementary waves of the exact Riemann solution in Test 4, see Fig n q u p a UL U1 U2 U3 U4 U5 UR 1.3 1.872903 3.775791 2.969906 0.533582 2.363115 1.775738 0.880818 1.250641 3.067818 3.067818 3.675948 1.66725 4.643562 3.318027 0.3 0.3 1 0.78177 0.78177 0.7 0.7 0.7 0.7 Fig Test The configuration of the exact Riemann solution in the ðx; tÞ-plane The states U ; U , and U are distributed along the t-axis Riemann solution are given by Table 17 in [33], where the authors compare the exact Riemann solution with approximate solutions obtained from the one-dimensional and the three-dimensional models For the sake of completeness, we list these states in Table The exact Riemann solution starts by a stationary wave from U L to U , followed by a 1-shock with zero speed from U to U , then followed by another stationary wave from U to U It is attached by a rarefaction wave from U to U , and it contiues with a 2-contact discontinuity from U to U , and finally it reaches U R by a 3-shock See Fig 5, where the states U ; U , and U are distributed along the t-axis in the ðx; tÞ-plane Our scheme using Corrector (II) in (4.7) computes the approximate solution at the time t ¼ 0:2s over the interval ½0; 2 with 6000 mesh points As in [33], the initial discontinuity is located at x ¼ 0:8 The density and velocity of the exact and approximate solutions are displayed in Fig Fig shows that the scheme can provides us with a good approximation to the exact solution Table States that separate the elementary waves of the exact Riemann solution in Test 5, see Fig UL U1 U2 U3 U4 U5 UR q u p a 2.363115 0.533582 2.969906 3.775791 1.872903 1.3 À3.675948 À3.067818 À3.067818 À1.250641 À0.880818 À1.775738 À2 0.3 0.3 3.318027 4.643562 1.66725 0.7 0.7 0.7 0.7 0.78177 0.78177 Fig Test The configuration of the exact Riemann solution in the ðx; tÞ-plane The states U ; U , and U are distributed along the t-axis 5.6 Test We consider the approximation of an exact Riemann solution that may also contain three waves of the same zero speed The states that determine the elementary waves of the exact Riemann solution are given by Table 16 in [33], where the authors compare the exact Riemann solution with approximate solutions obtained from the one-dimensional and the three-dimensional models Precisely, these states are given in Table The exact Riemann solution starts by a 1-rarefaction wave from U L to U , followed by a 2-contact discontinuity from U to U , then followed by a rarefaction wave from U to U It is attached by a stationary wave from U to U , followed by a 1-shock wave with zero speed from U to U , and finally it reaches U R by another stationary wave See Fig 7, where the states U ; U , and U are distributed along the t-axis in the ðx; tÞ-plane Fig Test Comparison of the the density and velocity of the exact solution and the ones of the approximate solution by the well-balanced scheme equipped with Corrector (II) in (4.7) over ½0; 2 at the time t ¼ 0:2s 138 M.D Thanh, D Krưner / Computers & Fluids 66 (2012) 130–139 Fig Test Comparison of the the density and velocity of the exact solution and the ones of the approximate solution by the well-balanced scheme equipped with Corrector (II) in (4.7) over ½0; 2 at the time t ¼ 0:12s Table States that separate the elementary waves of the exact Riemann solution in Test 6, see Fig UL U1 U2 U3 U4 U5 UR q u p a 1.4 2.077419 0.3 0.258647 0.200487 0.232799 0.269081 À2 À2.591083 À2.591083 À3.379023 À4.359267 À3.837976 À3.273959 3.5 3.5 2.84374 1.98703 2.449388 1 0.889412 0.889412 0.87 0.87 Fig Test The configuration of the exact Riemann solution in the ðx; tÞ-plane The states U ; U , and U are distributed along the t-axis Our scheme using Corrector (II) in (4.7) computes the approximate solution at the time t ¼ 0:12s over the interval ½0; 2 with 6000 mesh points As in [33], the initial discontinuity is located at x ¼ 0:8 The density and velocity of the exact and approximate solutions are displayed in Fig 8, where we display the solutions over the interval ½0; 1:2 for a better view Fig shows that the scheme can give a good approximation to the exact solution 5.7 Test let us consider another exact Riemann solution given by Table 16 in [33], where the authors compare the exact Riemann solution with approximate solutions obtained from the one-dimensional and the three-dimensional models The states that determine the elementary waves of the exact Riemann solutions are given in Table The exact Riemann solution starts by a 1-shock wave from U L to U , followed by a 2-contact discontinuity from U to U , then followed by a stationary wave from U to U It continues with a 1-shock with zero speed from U to U , followed by another stationary wave from U to U , and finally it reaches U R by a 3-rarefaction wave See Fig 9, where the states U ; U , and U are distributed along the t-axis in the ðx; tÞ-plane Our scheme using Corrector (II) in (4.7) computes the approximate solution at the time t ¼ 0:15s over the interval ½0; 2 with 6000 mesh points As in [33], the initial discontinuity is located at x ¼ 0:8 The density and velocity of the exact and approximate solutions are displayed in Fig 10 in the interval ½0; 1:5 Fig 10 also Fig 10 Test Comparison of the the density and velocity of the exact solution and the ones of the approximate solution by the well-balanced scheme equipped with Corrector (II) in (4.7) over ½0; 2 at the time t ¼ 0:15s M.D Thanh, D Krưner / Computers & Fluids 66 (2012) 130–139 indicates that our scheme can provide a reasonable approximation to the exact solution Conclusions Most existing schemes for nonconservative systems or nonstrictly hyperbolic systems can approximate the exact solutions only in strictly hyperbolic domains This work gives a way to treat numerically the nonconservative terms of the model of a fluid in a nozzle with variable cross-section in the resonant regime where data belong to both sides of the resonant surface We introduce two types of computing correctors to ‘‘navigate’’ the scheme to take the right state Tests show that our well-balanced method equipped by one of these computing correctors gives good approximations Questions on a general approach and higher-order schemes are open for further study Acknowledgments The authors are grateful to the reviewers for their very constructive comments and helpful suggestions This research is funded by Viet Nam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 101.02-2011.36 References [1] Ambroso A, Chalons C, Coquel F, Galié T Relaxation and numerical approximation of a two-fluid two-pressure diphasic model ESAIM: M2AN 2009;43:1063–97 [2] Andrianov N, Warnecke G On the solution to the Riemann problem for the compressible duct flow SIAM J Appl Math 2004;64:878–901 [3] Andrianov N, Warnecke G The Riemann problem for the Baer–Nunziato model of two-phase flows J Comput Phys 2004;195:434–64 [4] Audusse E, Bouchut F, Bristeau M-O, Klein R, Perthame B A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows SIAM J Sci Comput 2004;25:2050–65 [5] Bouchut F Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources Frontiers in Mathematics series, Birkhäuser; 2004 [6] Botchorishvili R, Perthame B, Vasseur A Equilibrium schemes for scalar conservation laws with stiff sources Math Comput 2003;72:131–57 [7] Botchorishvili R, Pironneau O Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws J Comput Phys 2003;187:391–427 [8] Chinnayya A, LeRoux A-Y, Seguin N A well-balanced numerical scheme for the approximation of the shallow water equations with topography: the resonance phenomenon Int J Finite 2004;1(4) [9] Castro MJ, Gallardo JM, Parés C High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products Appl Shallow-Water Sys Math Comput 2006;75:1103–34 [10] Castro MJ, LeFloch PG, Muñoz-Ruiz ML, Parés C Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes J Comput Phys 2008;227:8107–29 [11] Dal Maso G, LeFloch PG, Murat F Definition and weak stability of nonconservative products J Math Pures Appl 1995;74:483–548 [12] Dumbser M, Castro M, Parés C, Toro EF ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows Comput Fluids 2009;38:1731–48 [13] Dumbser M, Hidalgo A, Castro M, Parés C, Toro EF FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems Comput Methods Appl Mech Eng 2010;199:625–47 139 [14] Gallouët T, Hérard J-M, Seguin N Numerical modeling of two-phase flows using the two-fluid two-pressure approach Math Models Methods Appl Sci 2004;14:663–700 [15] Goatin P, LeFloch PG The Riemann problem for a class of resonant nonlinear systems of balance laws Ann Inst H Poincar Anal NonLinéaire 2004;21:881–902 [16] Gosse L A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms Comput Math Appl 2000;39:135–59 [17] Greenberg JM, Leroux AY A well-balanced scheme for the numerical processing of source terms in hyperbolic equations SIAM J Numer Anal 1996;33:1–16 [18] Greenberg JM, Leroux AY, Baraille R, Noussair A Analysis and approximation of conservation laws with source terms SIAM J Numer Anal 1997;34:1980–2007 [19] Isaacson E, Temple B Nonlinear resonance in systems of conservation laws SIAM J Appl Math 1992;52:1260–78 [20] Isaacson E, Temple B Convergence of the  godunov method for a general resonant nonlinear balance law SIAM J Appl Math 1995;55:625–40 [21] Jin S, Wen X An efficient method for computing hyperbolic systems with gometrical source terms having concentrations J Comput Math 2004;22:230–49 [22] Jin S, Wen X Two interface type numerical methods for computing hyperbolic systems with gometrical source terms having concentrations SIAM J Sci Comput 2005;26:2079–101 [23] Kröner D, LeFloch PG, Thanh MD The minimum entropy principle for fluid flows in a nozzle with discntinuous crosssection ESAIM: M2AN 2008;42:425–42 [24] Kröner D, Thanh MD Numerical solutions to compressible flows in a nozzle with variable cross-section SIAM J Numer Anal 2005;43:796–824 [25] Lallemand M-H, Saurel R Pressure relaxation procedures for multiphase compressible flows INRIA Report, No 4038; 2000 [26] LeFloch PG Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form Comm Part Diff Equa 1988;13:669–727 [27] LeFloch PG Shock waves for nonlinear hyperbolic systems in nonconservative form Institute for Math and its Appl., Minneapolis, Preprint# 593; 1989 (unpublished) [28] LeFloch PG, Thanh MD The Riemann problem for fluid flows in a nozzle with discontinuous cross-section Comm Math Sci 2003;1:763–97 [29] LeFloch PG, Thanh MD The Riemann problem for shallow water equations with discontinuous topography Comm Math Sci 2007;5:865–85 [30] LeFloch PG, Thanh MD A godunov-type method for the shallow water equations with variable topography in the resonant regime J Comput Phys 2011;230:7631–60 [31] Marchesin D, Paes-Leme PJ A Riemann problem in gas dynamics with bifurcation Hyperbolic partial differential equations III Comput Math Appl (Part A) 1986;12:433–55 [32] Parés C Numerical methods for nonconservative hyperbolic systems: a theoretical framework SIAM J Numer Anal 2006;44:300–21 [33] Rochette D, Clain S, Bussiére W Unsteady compressible flow in ducts with varying cross-section: comparison between the nonconservative Euler system and the axisymmetric flow model Comput Fluids 2012;53:53–78 [34] Rosatti G, Begnudelli L The Riemann problem for the one-dimensional freesurface shallow water equations with a bed step: theoretical analysis and numerical simulations J Comput Phys 2010;229:760–87 [35] Rhebergen S, Bokhove O, van der Vegt JJW Discontinuous galerkin finite element methods for hyperbolic nonconservative partial differential equations J Comput Phys 2008;227:1887–922 [36] Saurel R, Abgrall R A multi-phase godunov method for compressible multifluid and multiphase flows J Comput Phys 1999;150:425–67 [37] Toumi I A weak formulation of Roe’s approximate Riemann solver J Comput Phys 1992;102:360–73 [38] Thanh MD The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area SIAM J Appl Math 2009;69:1501–19 [39] Thanh MD, Fazlul MD K, Ismail A Izani MD Well-balanced scheme for shallow water equations with arbitrary topography, Inter J Dyn Sys Diff Eqs 2008;1:196–204 [40] Thanh MD, Ismail A Izani Md Well-balanced scheme for a one-pressure model of two-phase flows Phys Scr 2009;79 [41] Thanh MD, Kröner D, Nam NT Numerical approximation for a Baer–Nunziato model of two-phase flows Appl Numer Math 2011;61:702–21 [42] Thanh MD On a two-fluid model of two-phase compressible flows and its numerical approximation Comm Nonl Sci Num Simul 2012;17:195–211 ... by a stationary contact that is attached to it at U and jumps into U 2 G1 The attaching condition means that the characteristic speed k1 ðU Þ at the end of the rarefaction fan coincides with. .. to both sides of the resonant surface and that a rarefaction wave in one side is attached by a stationary wave that jumps into the other side we can rewrite the resonant surface Rỵ as Rỵ : u ẳ... the nonconservative terms of the model of a fluid in a nozzle with variable cross-section in the resonant regime where data belong to both sides of the resonant surface We introduce two types of