DSpace at VNU: Building fast well-balanced two-stage numerical schemes for a model of two-phase flows

Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Building fast well-balanced two-stage numerical schemes for a model of two-phase flows Mai Duc Thanh ⇑ Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam a r t i c l e i n f o a b s t r a c t Article history: Received 23 April 2013 Received in revised form 15 October 2013 Accepted 17 October 2013 Available online 30 October 2013 Keywords: Two-phase flow Well-balanced scheme Lax–Friedrichs scheme Richtmyer’s scheme Roe scheme We present a set of well-balanced two-stage schemes for an isentropic model of two-phase flows arisen from the modeling of deflagration-to-detonation transition in granular materials The first stage is to absorb the source term in nonconservative form into equilibria Then in the second stage, these equilibria will be composed into a numerical flux formed by using a convex combination of the numerical flux of a stable Lax–Friedrichs-type scheme and the one of a higher-order Richtmyer-type scheme Numerical schemes constructed in such a way are expected to get the interesting property: they are fast and stable Tests show that the method works out until the parameter takes on the value CFL, and so any value of the parameter between zero and this value is expected to work as well All the schemes in this family are shown to capture stationary waves and preserves the positivity of the volume fractions The special values of the parameter 0; 1=2; 1=1 ỵ CFLị, and CFL in this family define the Lax–Friedrichs-type, FAST1, FAST2, and FAST3 schemes, respectively These schemes are shown to give a desirable accuracy The errors and the CPU time of these schemes and the Roe-type scheme are calculated and compared The constructed schemes are shown to be well-balanced and faster than the Roe-type scheme Ó 2013 Elsevier B.V All rights reserved Introduction In this paper we aim to build a set of fast and well-balanced schemes for the following isentropic model of two-phase ows, @ t ag qg ị ỵ @ x ag qg ug ị ẳ 0; @ t ag qg ug ị ỵ @ x ag qg u2g ỵ pg ÞÞ ¼ pg @ x ag ; @ t ðas qs ị ỵ @ x as qs us ị ẳ 0; 1:1ị @ t as qs us ị ỵ @ x as qs u2s ỵ ps ịị ẳ pg @ x ag ; @ t qs ỵ @ x qs us ị ẳ 0; x R; t > 0: The model (1.1) is obtained from the two-phase mixture model to study the deflation-to-detonation transition in granular explosives, see [7], by simplifying the model An alternative form of the model was presented in [4], in which the compaction dynamics equation (the last equation of (1.1)) has a different form: @ t ag ỵ us @ x ag ẳ 0; Observe that the last equation involves a nonconservative term us @ x ag The reader is referred to [10] for the mathematical formulation of nonconservative hyperbolic systems involving nonconservative terms ⇑ Tel.: +84 2211 6965; fax: +84 3724 4271 E-mail address: mdthanh@hcmiu.edu.vn 1007-5704/$ - see front matter Ó 2013 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.cnsns.2013.10.017 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 1837 Throughout, we use the subscripts g and s to indicate the quantities in the g-phase (referred to as the gas phase), and in the solid phase (referred to as the solid phase), respectively However, our study in this work can be applied for more general materials For example, one of the two phases or both may be liquid The notations ak ; qk ; uk ; pk ; k ¼ g; s, stand for the volume fraction, density, velocity, and pressure in the k-phase, k ¼ g; s, respectively The volume fractions are constraint by the relation as ỵ ag ẳ 1: ð1:2Þ In our recent work [40], a Roe-type scheme was constructed by using the admissible solid contacts at each node to absorb the nonconservative terms The states on both sides of these solid contacts are incorporated into a Roe-type matrix of the decoupling system of (1.1) which is obtained from (1.1) by letting the volume fractions be constant A similar process could also be made by using another numerical flux, for example the one of the Lax–Friedrichs scheme to form a Lax–Friedrichstype scheme However, this Lax–Friedrichs-type scheme has less accuracy then the Roe-type scheme, as seen in Section This is probably because the Lax–Friedrichs scheme is, though stable, too diffusive Furthermore, the incorporation of the states on both sides of the solid contacts mentioned above into a higher-order scheme such as the Lax–Wendroff or Richmyer’s scheme does not yield satisfactory results: the scheme is numerically unstable, where large oscillations appear shortly in numerical tests Customarily, a well-balanced scheme is the one which can preserve the steady state solution exactly Motivated by the above argument, we aim to build in this paper a set of numerically stable schemes that can be faster and have a better accuracy than the Roe-type scheme For this purpose, we form a one-parameter family of numerical fluxes by using convex combinations of the numerical fluxes of the Lax–Friedrichs scheme and the second-order Richtmyer’s with a parameter h ẵ0; The states U njặ1;ầ on the other side of the solid contacts at xjỈ1=2 from any given states U njỈ1 are then incorporated in numerical fluxes in this family to produce new schemes The values h ẳ 0; 1=2; h ẳ 1=1 ỵ CFLị and CFL in this family define a Lax–Friedrichs-type, FAST1, FAST2, and FAST3 schemes, respectively Thus, the FAST1 and FAST2 schemes are formed in a similar way as the FORCE and GFORCE schemes Recall that the FORCE and GFORCE schemes are convex combinations of the Lax–Friedrichs scheme and Lax–Wendroff scheme, see [29,30,9] Tests of Lax–Friedrichs-type, FAST1, FAST2, and FAST3 schemes are presented Errors, order of convergence, numbers of iterations, CPU time are evaluated.All the tests show desirable approximations to the exact solutions The results are compared with a newly constructed Roe-type scheme [40] Tests show that the FAST3 scheme gives the better results than the Roe-type scheme Naturally, the same result is expected for schemes corresponding to the values of the parameter h closed to CFL, at least Moreover, we show that our schemes are well-balanced Observe that the restrictive attention to the isentropic case may cause a certain limitation for applications, and would motivate for future developments for the general case We note that numerical approximations of nonconservative systems have attracted attention of many authors Numerical well-balanced schemes for a single conservation law with a source term are presented in [15,16,5,6,3] Numerical schemes for multi-phase multi-pressure models were presented in [21,25,1,26,13,39,36] Various numerical schemes for two-fluid models of two-phase flows were constructed in [42,38,35] Well-balanced schemes for other nonconservative hyperbolic systems were built in [20,19,24,3,37,8,17].The Riemann problem for various nonconservative hyperbolic systems was studied in [22,33,23,14,32,2] Shock waves in two-fluid models of two-phase flows were studied in [18,34] Some recent Godunov-type schemes for various fluid flow models are presented in [31,27,24,28] See also the references therein The organization of this paper is as follows In Section we present basic concepts of the system (1.1): non-strict hyperbolicity, discontinuities, and admissible solid contact waves In Section we construct a one-parameter family of well-balanced schemes First, we describe a family of numerical fluxes by using convex combinations of the numerical fluxes of a stable Lax–Friedrichs and a higher-order Richmyer’s schemes Then, we show how to incorporate admissible solid contacts into these numerical fluxes to obtain a numerical scheme Section is devoted to numerical tests, where tests for the well-balanced Lax–Friedrichs-type scheme, FAST1, FAST2, FAST3, and Roe-type schemes are carried out Finally, in Section we draw several conclusions and discussions Preliminaries 2.1 Nonstrict hyperbolicity The governing Eqs (1.1) may be re-written as a system in nonconservative form: U t ỵ AUịU x ẳ 0; 2:1ị where qg Bu C B gC B C C U¼B B qs C; B C @ us A ag ug B B h0 q ị B g g B AUị ẳ B B B @ 0 qg 0 ug 0 qg ðug Àus Þ ag 0 us qs 0 hs ðqs Þ us pg Àps 0 us as qs C C C C C C C A ð2:2Þ 1838 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 and the ‘‘enthalpy in the isentropic case’’ hk is defined by dhk qk ị dpk qk ị ẳ ; dqk qk dqk k ¼ s; g: For example, in the case of an isentropic and ideal fluid, where the equation of state is given by p ¼ jqc ; j > 0; c > 1; ð2:3Þ the function hk is given by jk ck ck À1 q ; k ¼ s; g: ck k hk qk ị ẳ The matrix AðUÞ has five real eigenvalues sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi dpg dpg ; k2 Uị ẳ ug ỵ ; k1 Uị ¼ ug À dqg dqg sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi dps dps k3 Uị ẳ us ; k4 Uị ẳ us ỵ ; k5 Uị ẳ us ; dqs dqs 2:4ị provided that dpk > 0; dqk k ¼ s; g: It is interesting that the eigenvalues k5 may coincide with either k1 or k2 on a certain hyper-surface of the phase domain, called the sonic surface or resonant surface Due to the change of order of these eigenvalues when they are real, we set G1 G2 G3 Cỵ C :ẳ fU j k1 Uị > k5 Uịg; :ẳ fU j k1 Uị < k5 Uị < k2 Uịg; :ẳ fU j k2 Uị < k5 Uịg; :ẳ fU j k1 Uị ¼ k5 ðUÞg; :¼ fU j k2 ðUÞ ¼ k5 ðUÞg: Providing that dpg dqg ð2:5Þ > 0, we can define the supersonic region to be the one in which sffiffiffiffiffiffiffiffi dpg ; jug À us j > cg :¼ dqg which is G1 [ G3 The subsonic region is the one in which jug À us j < cg ; which is G2 See Fig The system thus fails to be strictly hyperbolic on the sonic surface C :ẳ Cỵ [ C : 2:6ị In each region Gi ; i ¼ 1; 2; 3, the system is strictly hyperbolic 2.2 Discontinuities Given a discontinuity of (1.1) of the form Ux; tị ẳ  U ; for x < rt; Uỵ ; for x > rt; 2:7ị where U Ỉ are the left-hand and right-hand states, and r is the speed of discontinuity propagation The usual jump relations for this discontinuity by the two conservative equations of mass conservation and compaction dynamics are given by À r½as qs ỵ ẵas qs us ẳ 0; rẵqs ỵ ẵqs us ẳ 0; 2:8ị where ẵA ẳ Aỵ A , and Aặ denote the values on the right and left of the jump of the quantity A The Eq (2.8) can be rewritten as ẵas qs us rị ẳ 0; ẵqs us rị ẳ 0; M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 1839 Fig The phase domain in the the plane us  constant or qs ðus rị ẳ M ẳ constant; Mẵas ẳ 0: ð2:9Þ The second equation of (2.9) implies that either M ¼ or ½as Š ¼ Since qs > 0, one obtains the following conclusion: across any discontinuity (2.7) of (1.1) À either us ¼ r ¼ constant; or ẵas ẳ 0: 2:10ị 2.3 Admissible solid contacts Let U ¼ ðqg0 ; ug0 ; qs0 ; us0 ; ag0 Þt be a given and fixed state We look for any state U ¼ ðqg ; ug ; qs ; us ; ag Þt that can be connected with U by a discontinuity where us  constant, as seen by (2.10) In this case, the discontinuity was shown to be associated with the 5th characteristic field, and so it is referred to as a 5-contact wave, or a solid contact, see [11,2,39,32] Moreover, the state U satisfies the equations us ¼ us0 ¼ r; ag qg ðug us ị ẳ ag0 qg0 ug0 us ị :ẳ m; 2:11ị ug us ị ỵ 2hg ẳ ug0 us0 ị ỵ 2hg0 and ps ¼ as0 ps0 À mðug À ug0 Þ À ðag pg À ag0 pg0 Þ : as ð2:12Þ ag ; qg and ug can be found using (2.11) The solid pressure is given by (2.12), and therefore the solid density can be calculated by using an equation of state of the form qs ẳ qs ps ị Thus, the state corresponding state U ¼ ðqg ; ug ; qs ; us ; ag Þt on the other side of the solid contact can be completely determined by (2.11) and (2.12) If we choose the left-hand state to be U À ¼ U , then the corresponding right-hand state is given by U ỵ ẳ U satisfying (2.11) and (2.12); if we choose the right-hand state to be U ỵ ¼ U , then the corresponding left-hand state is given by U À ¼ U satisfying (2.11) and (2.12) These states U Ỉ will determine the points qg;Ỉ B q ug;Ỉ C C B g;Ỉ V Ỉ :ẳ B C; @ qs;ặ A 2:13ị qs;ặ us;ặ on both sides of the solid contact, which will be used in the construction of well-balanced schemes in the next section For simplicity, in the rest of this subsection we will drop the subindex ‘‘g’’ for the quantities in the gas phase Moreover, we assume that the fluid in the gas phase is isentropic and ideal, that is, the equation of state on the gas phase is of the form (2.3) Then, it follows from (2.11) that the gas density satisfies the following nonlinear algebraic equation   WðU ; a; qị :ẳ lqc u0 us ị2 ỵ lqc01 q ỵ where lẳ 2jc : c1  a0 u0 us ịq0 a 2 q ẳ 0; q > 0; ð2:14Þ 1840 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 The function W in (2.14) is strictly convex It possesses two zeros whenever a P amin U ị :ẳ p jc a0 q0 ju0 us j lcỵ1ị u0 us ị2 ỵ cỵ1 qc01 2:15ị : 2ccỵ1 1ị It is natural to impose an admissibility condition to select the meaningful solution Following [22,33,32], we impose the following admissibility criterion for contact discontinuities (MC) Any contact wave does not cross the resonant surface C It is not difficult to verify that WðU ; a; qÞ > 0; WðU ; a; qÞ > 0; a0 ðu0 À us Þq0 ffi; for q qà :¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ðu0 À us ị2 ỵ lqc01 for q P q :ẳ u0 us ị2 l 2:16ị !1=c1ị c1 ỵ q0 : The admissible zero can be determined and calculated as follows (i) For U G1 [ G3 , the admissible value of the density is the smaller zero q ¼ u1 ðU ; aÞ of W, which can be computed using Newton’s method starting at qà defined by (2.16); (ii) For U G2 , the admissible value of the density is the larger zero q ¼ u2 ðU ; aÞ of W, which can be computed using Newton’s method starting at qà defined by (2.16) A set of well-balanced schemes 3.1 The underlying numerical fluxes for the decoupling system It is derived from (2.10) that if ẵas ẳ 0, then the volume fractions remain constant across the discontinuity The governing equations of the system (1.1) is therefore reduced to the following decoupling system of two independent sets of isentropic gas dynamics equations in both phases @ t V ỵ @ x f Vị ẳ 0; ð3:1Þ where qg C B B qg ug C C B V ¼B C; B q C @ s A qs us qg ug C B B q u2 ỵ pg C C B g g f Vị ẳ B C: B q us C A @ s 3:2ị qs u2s ỵ ps Waves of the system of conservation laws (3.1) associate with the four characteristic elds ki ; r i ị; i ẳ 1; 2; 3; 4, where r i is a right-eigenvector of the matrix A given by (2.2) corresponding to ki ; i ¼ 1; 2; 3; 4, respectively Denote by Dt the time step, and by Dx the spacial mesh size, which satisfy the following C.F.L stability condition  qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi CFL ẳ kmax jug j ỵ p0g qg ị; jus j ỵ p0s qs ị < 1; U Dt : Dx 3:3ị n ẳ 0; 1; 2; ; 3:4ị k :ẳ Explicit schemes for (3.1) are given by   V nỵ1 ẳ V nj k gV njỵ1 ; V nj ị gV nj ; V njÀ1 Þ ; j j Z; where gðU; VÞ is the numerical flux The CFL number is given by CFL ẳ Dt maxfjki Vịj; i ẳ 1; 2; 3; 4g < 1: Dx V ð3:5Þ The rest of this subsection involves the Lax–Friedrichs scheme, Richtmyer’s scheme, and Roe’s scheme (see[12], for example) M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 1841 Lax–Friedrichs scheme for the decoupling system The numerical flux of the Lax–Friedrichs scheme is given by g LF U; Vị ẳ 1 f Uị ỵ f Vịị V Uị: 2k ð3:6Þ Richtmyer’s scheme for the decoupling system Richtmyer’s scheme is a second-order scheme that has the numerical flux  g R U; Vị ẳ f  UỵV k ðf ðVÞ À f ðUÞÞ : 2 ð3:7Þ Convex combinations the numerical fluxes of the Lax–Friedrichs scheme and Richtmyer’s scheme These schemes are constructed by using convex combinations of the Lax–Friedrichs scheme and Richtmyer’s scheme Precisely, let us be given the Lax–Friedrichs numerical flux g LF ðU; VÞ and the Richtmyer numerical flux g R ðU; VÞ We define g h U; Vị :ẳ hịg LF U; Vị ỵ hg R U; Vị; h 1; ð3:8Þ where g LF and g R are given by (3.6) and (3.7), respectively A Roe matrix for the decoupling system We dene p p qu ỵ qu  ẳ pLL pR R ; u qL ỵ qR (p R ÀpL qL – qR ; if qL ¼ qR if qR qL ; ẳ p p qL ị; and the notations hold for the corresponding quantities in both phases A Roe scheme for the decoupling system (3.1) is given by 0 B B ug ỵ p g B AU L ; U R ị ẳ B B @ g 2u 0 0 s us ỵ p 0 C C C C; C A ð3:9Þ s 2u The Roe matrix given by (3.9) has four eigenvalues pffiffiffiffiffi p k1 U L ; U R ị ẳ u g p g ỵ p g ; k2 U L ; U R ị ẳ u g ; p p s p s ỵ p s ; k4 U L ; U R ị ẳ u s : k3 U L ; U R ị ẳ u The corresponding right-eigenvectors can be chosen as 1 C B B k1 ðU L ; U R Þ C C r U L ; U R ị ẳ B C; B C B A @ 1 C B B k2 ðU L ; U R Þ C C r ðU L ; U R Þ ¼ B C B C B A @ 0 and B B r ðU L ; U R ị ẳ B B B @ 0 k3 ðU L ; U R Þ C C C C; C A B B r U L ; U R ị ẳ B B B @ 0 1 C C C C: C A k4 ðU L ; U R Þ The corresponding left-eigenvectors of the Roe matrix can be chosen as 1842 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 À1 l1 ðU L ; U R Þ ¼ p ffiffiffiffiffi ðÀk2 ðU L ; U R Þ; 1Þ; g p À1 l2 ðU L ; U R ị ẳ p k1 U L ; U R Þ; 1Þ; g p l ðU ; U Þ ¼ À1 pffiffiffiffiffi ðÀk4 ðU L ; U R Þ; 1Þ; L R s p l4 ðU L ; U R ị ẳ p k3 U L ; U R Þ; 1Þ: s p Define the coefcients U L ; U R ị ẳ li ðU L ; U R ÞðU R À U L ị; i ẳ 1; 2; 3; 4: A Roe numerical flux for the system (3.1) is given by g Roe V; Wị ẳ 1X FVị ỵ FWịị jki jai r i ịV; Wị; 2 iẳ1 ð3:10Þ where V; FðVÞ are given by (3.2) 3.2 Building two-stage well-balanced schemes It follows from the equation of conservation of mass in the solid phase and the compaction dynamics equation that @ t ag ỵ us @ x ag ¼ 0: ð3:11Þ When the volume fraction ag changes across the solid contact waves, the solid velocity us still remains constant This suggests that we can apply an upwind scheme technique for this equation: n anỵ1 g;j ẳ ag;j  Dt  ỵ;n n n n us;j ag;j ang;j1 ị ỵ u;n s;j ag;jỵ1 ag;j ị ; Dx 3:12ị where uỵ ẳ maxfu; 0g; u ẳ minfu; 0g: Next, we define the admissible solid contact at each node x ẳ xjỵ1=2 ẳ j ỵ 1=2ịDx; t ẳ tn ¼ nDt; j Z; n ¼ 0; 1; 2;  The volume fraction change across the node x ẳ xjỵ1=2 ẳ j ỵ 1=2ịDx; t ¼ tn ¼ nDt, creates a 5-contact discontinuity jumping from the given state V njỵ1 with the gas volume fraction anjỵ1 to an admissible state denoted by V njỵ1; with the gas volume fraction anj ;  The volume fraction change across the node x ¼ xjÀ1=2 ¼ ðj À 1=2ịDx; t ẳ t n ẳ nDt, creates a 5-contact discontinuity jumping from the given state V njÀ1 with the gas volume fraction anjÀ1 to an admissible state denoted by V nj1;ỵ with the gas volume fraction anj The corresponding states V njặ1;ầ are of the form (2.13), and can be computed using the strategy described in the Section 2.3 Moreover, to make sure that the process of computing the admissible solid contacts always works, we suggest an additional computing technique For example, if the gas phase is ideal, we can substitute the volume fraction ang;j by its ‘‘relaxed’’ value defined by n o an;Relax ¼ max ang;j ; amin ang;jặ1 ; qng;jặ1 ; ung;jặ1 ị ; g;j 3:13ị when computing V njặ1;ầ , respectively, where amin is dened by (2.15) A one-parameter family of numerical schemes for (1.1) is constructed as follows The state V njỵ1; is substituted for V njỵ1 , and the state V nj1;ỵ is substituted for V njÀ1 into the scheme (3.4), respectively, j Z; n ¼ 0; 1; 2; , where the numerical flux g ¼ g h is defined by (3.8) Together with (3.12), this yields a one-parameter family of schemes dened by   ỵ;n ;n nỵ1 n Dt n n n n > < ag;j ¼ ag;j Dx us;j ag;j ag;j1 ị ỵ us;j ag;jỵ1 ag;j ị ;   > : V nỵ1 ẳ V n Dt g V n ; V n Þ À g ðV n ; V n Þ ; j Z; h h j j j jỵ1; j1;ỵ j Dx where h ẵ0; plays the role of a parameter, and 3:14ị n ẳ 0; 1; 2; 1843 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 qg Bq u C gC B V ¼ B g C: @ qs A qs us Table Test – The equilibrium states Quantitiesn States UL UR ag 0.5 0.4 3.0314331 0.6 0.251465 1.1609305 3.7109308 pg ug ps us Fig Test 1: The exact stationary contact wave in the ðx; tÞ-plane (upper-left corner), and its approximations at the time t ¼ 0:1 in the interval x ½À1; 1Š with 1000 mesh points 1844 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Taking h ¼ in (3.14), one obtains a Lax–Friedrichs-type (LF-type) scheme We define the FAST1, FAST2and FAST3 schemes to be the schemes defined by (3.14) by taking h ¼ 0; 1=2; 1=1 ỵ CFLị, and CFL, respectively A Roe-type scheme was constructed in [40] by   ỵ;n ;n n Dt n n n n > < anỵ1 g;j ẳ ag;j Dx us;j ag;j ag;j1 ị ỵ us;j ag;jỵ1 ag;j ị ;   > : V nỵ1 ẳ V nj DDxt g Roe V nj ; V njỵ1; ị g Roe V nj1;ỵ ; V nj Þ ; j ð3:15Þ where g Roe is given by (3.10) The schemes (3.14) possesses nice properties as in the following theorem Proposition 3.1 The following conclusions hold (i) (Well-balanced scheme)The schemes (3.14) capture exactly steady state solutions This means that for any stationary contact wave, it holds that U nỵ1 ẳ U nj ; j j Z; n ẳ 0; 1; 2; ::; 3:16ị where U ¼ ðqg ; ug ; qs ; us ; ag Þ Thus, the schemes (3.14) are well-balanced (ii) (Positivity of volume fractions) The well-balanced schemes (3.14) preserve the positivity of the volume fractions This means that if a0k;j > for all j Z, then ank;j > for all j Z; n ¼ 1; 2; 3; ; k ¼ s; g Proof (i) Let a steady state solution, which is in this case a stationary contact discontinuity, be given, where us ¼ It follows from (3.12) that the volume fraction in the gas phase remain unchanged: n anỵ1 j Z; n ¼ 0; 1; 2; g;j ¼ ag;j ; ð3:17Þ and so does the volume fraction in the solid phase Furthermore, it is derived from the relations (2.11) that qng;jỵ1; ẳ qng;j ; qng;j1;ỵ ẳ qng;j ; qns;jỵ1; ẳ qns;j ; qns;j1;ỵ ẳ qns;j ; ung;jỵ1; ẳ ung;j ; ung;j1;ỵ ẳ ung;j ; uns;jỵ1; ẳ uns;j ; uns;j1;ỵ ẳ uns;j : This means that V njỵ1; ẳ V nj ; V nj1;ỵ ẳ V nj : The last equations yield g h V nj ; V njỵ1; ị g h V nj1;ỵ ; V nj ị ẳ g h ðV nj ; V nj Þ À g h V nj ; V nj ị ẳ 0: Table Test – The initial values Quantitiesn States UL UR ag 0.5 0.052089093 À4.2053278 0.69975173 0.27397458 0.8 0.077244506 À4.0462008 1.6366822 À0.32129324 pg ug ps us Table Test – The states that separate the elementary waves of the exact Riemann solution n U1 U2 U3 U4 ag 0.5 0.071190298 À4.3333333 0.69975173 0.27397458 0.5 0.15157166 À4 0.69975173 0.27397458 0.5 0.15157166 À4 0.8 0.077244506 À4.0462008 2.3389469 pg ug ps us M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 1845 It therefore follows from (3.14) that V nỵ1 ẳ V nj ; j j Z; n ẳ 0; 1; 2; 3:18ị From (3.17) and (3.18) we obtain (3.16) The proof of (ii) is omitted, since it is similar to the one of Theorem 4.1 in [36] Proposition 3.1 is completely proved h Numerical tests This section is devoted to numerical tests, where approximate solutions obtained by the schemes (3.14) are compared with the exact solution of the Riemann problem for (1.1) For these tests, both phases are assumed to have an equation of state of the form (2.3) The exact solution of the Riemann problem consists of elementary waves: shocks, rarefaction waves, Fig Test Approximate solution by the two-stage method using different underlying numerical fluxes: Lax–Friedrichs, Roe, and FAST3 with 250 mesh points 1846 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Table Test – Errors, orders of convergence, numbers of iterations, and CPU time of the Lax–Friedrichs-type, FAST1, FAST2, Roe-type, and FAST3 schemes LF-type N L1 -Error Order Iterations CPU time 250 500 1000 2000 4000 0.10292 0.070595 0.04696 0.030349 0.01881 – 0.54 0.59 0.63 0.69 75 151 304 611 1223 5.07 12.246 34.227 114.77 512.04 FAST1 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.072043 0.0468 0.030291 0.018785 0.011097 – 0.62 0.63 0.69 0.76 76 152 305 612 1225 7.4412 19.656 61.527 229.13 952.25 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.066133 0.042573 0.027533 0.01678 0.0097982 – 0.64 0.63 0.71 0.78 76 152 306 612 1225 7.4568 19.594 61.293 228.9 943.23 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.056066 0.03609 0.022792 0.013602 0.007675 – 0.64 0.66 0.74 0.83 244 490 983 1968 3938 22.246 60.622 190.68 707.87 2977.7 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.048951 0.030578 0.019017 0.011169 0.0062571 – 0.68 0.69 0.77 0.84 76 153 306 613 1226 7.2072 19.344 57.845 232.77 955.19 FAST2 Roe-type FAST3 and a solid contact As seen above, the volume fractions change only across the solid contact, and the shocks and rarefaction waves behave as in the usual gas dynamics equations of each individual phase Therefore, we will make five numerical tests relying on the location and the sign of the solid velocity of the solid contact in the exact Riemann solution The parameters for Tests 1–4 are jg ¼ 1; js ¼ 0:4; cg ¼ 1:4; cs ¼ 1:6; while the parameters for dioxygen and water at the temperature 100 °C are taken for Test 4.1 Numerical test 1: stationary contact discontinuities This test will verify the well-balanced property of the proposed family of schemes The approximate solution will be computed at the time t ¼ 0:1 on the interval ½À1; 1Š of the x-space with 1000 mesh points We take the underlying numerical flux to be the one of the LF-type scheme with CFL = 3/4 Consider the Riemann problem for (1.1) and (1.2) with the initial data given in Table It is not difficult to check that in this case the Riemann solution is a stationary 5th contact wave The Fig shows that the stationary contact wave is well captured 4.2 Numerical test 2: solid contact in supersonic region with zero velocity In this test, we consider the Riemann problem for (1.1) and (1.2) with the initial data given in Table The volume fraction jump is large: jaL À aR j ¼ 0:5 The Riemann solution has a stationary contact discontinuity The exact Riemann solution begins with a 1-shock wave from U L to U This shock is followed by a 2-rarefaction wave from U to U , followed by a 3-shock 1847 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Fig Test Approximate solutionusing the underlying FAST3 numerical flux The approximate solution is computed at the time t ¼ 0:1 on the interval ½À1; 1Š with 250, 1000, and 4000 mesh points, and is compared with the exact solution Table Test – The initial values Quantitiesn States UL UR ag 0.5 0.77237457 1.1881594 0.84486635 3.063069 0.52 1.5691914 1.2538958 4.1965469 0.7166676 pg ug ps us 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 These states are given in Table The velocity of the solid contact between U and U is equal to zero It is not difficult to check 1848 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Table Test – The states that separate the elementary waves of the exact Riemann solution n U1 U2 U3 U4 ag 0.5 1.0556063 0.84486635 3.063069 0.5 1.0556063 5.7995461 1.1 0.52 1.0563705 1.0038958 5.9971941 1.1 0.52 1.5691914 1.2538958 5.9971941 1.1 pg ug ps us Fig Test Approximate solution by the two-stage method using different underlying numerical fluxes: Lax–Friedrichs, Roe, and FAST3 with 250 mesh points that all these states, and in particular the states on both sides of the solid contact, belong to the supersonic region G3 The structure of the Riemann solution is shown in Fig (upper-left corner) 1849 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Table Test – Errors, orders of convergence, numbers of iterations, and CPU time of the Lax–Friedrichs-type, FAST1, FAST2, Roe-type, and FAST3 schemes LF-type N L1 -Error Order Iterations CPU time 250 500 1000 2000 4000 0.33862 0.20095 0.11948 0.07918 0.053008 – 0.75 0.75 0.59 0.58 67 133 266 531 1062 4.5864 10.967 30.732 106.53 473.95 FAST1 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.20667 0.12402 0.076152 0.054041 0.040526 – 0.74 0.7 0.49 0.42 67 133 266 531 1062 7.2852 18.127 56.566 202.99 856.46 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.18667 0.11249 0.069365 0.050438 0.0387 – 0.73 0.7 0.46 0.38 67 133 266 531 1062 7.2072 18.455 57.908 204.14 853.75 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.18872 0.11909 0.078208 0.059893 0.045437 – 0.66 0.61 0.38 0.4 215 429 858 1716 3431 20.764 57.518 174.35 651.23 2676.9 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.13558 0.082602 0.051525 0.040979 0.034157 – 0.71 0.68 0.33 0.26 67 133 266 532 1063 6.864 18.845 58.953 205.86 856.74 FAST2 Roe-type FAST3 The approximate solution will be computed at the time t ẳ 0:1 on the interval ẵ1; of the x-space using each of the LFtype, FAST1, FAST2and FAST3 schemes, and the Roe-type scheme (3.15) with different mesh sizes Recall that the LF-type, FAST1, FAST2and FAST3 schemes are the ones given by (3.14) with the choice h ¼ 0; 1=2; 1=1 ỵ CFLị, and CFL, respectively The CFL number for LF-type (LF-type), FAST1, FAST2, and FAST3 can be taken by 3/4 However, the CFL number for the Roetype scheme is just 1/4, as it may not work for larger values, for example CFL = 1/3 The errors in the L1 norm are computed and the orders of convergence are estimated The errors, orders of convergence, the number of iterations, and the CPU time are reported by Table Fig shows the plots of the approximate solutions given by the LF-type, Roe-type, and FAST3 schemes Fig displays the approximate solutions by the FAST3 scheme with different mesh sizes As shown by Table 4, the accuracy of the Roe-type scheme is better than the LF-type, FAST1, and FAST2 schemes However, the CPU time of the Roe-type scheme is larger than the others The FAST3 scheme gives the best results, where it provides better errors with a smaller CPU time 4.3 Numerical test 3: solid contact in subsonic region with positive velocity In this test, we consider the Riemann problem for (1.1) and (1.2) with the initial data given in Table The exact Riemann solution begins with a 1-shock wave from U L to U This shock is followed by a 3-shock wave from U to U , followed by a solid contact from U to U , followed by a 2-rarefaction wave from U to U and then followed by a 4-shock wave from U to U R These states are given in Table It is easy to see that the states U ; U on both sides of the solid contact are located in the subsonic region, where the velocity of the solid contact is positive (us ¼ 1:1) The structure of the Riemann solution is shown in Fig (upper-left corner) The approximate solution will be computed at the time t ¼ 0:1 on the interval ½À1; 1Š of the x-space using each of the LF-type, FAST1, FAST2and FAST3 schemes, and the Roe-type scheme (3.15) with different mesh sizes.The CFL number for 1850 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Fig Test Approximate solutionusing the underlying FAST3 numerical flux The approximate solution is computed at the time t ¼ 0:1 on the interval ½À1; 1Š with 250, 1000, and 4000 mesh points, and is compared with the exact solution Table Test – The initial values Quantitiesn States UL UR ag 0.5 0.02336948 À3.2191769 0.23083198 0.12253651 0.45 0.086480129 À2.9865518 0.21493885 À0.31957976 pg ug ps us 1851 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Table Test – The states that separate the elementary waves of the exact Riemann solution n U1 U2 U3 U4 ag 0.5 0.031939128 À3.3333333 0.23083198 0.12253651 0.5 0.074136102 À3 0.23083198 0.12253651 0.5 0.074136102 À3 0.32987698 À0.1 0.45 0.086480129 À2.9865518 0.30716444 À0.1 pg ug ps us Fig Test Approximate solution by the two-stage method using different underlying numerical fluxes: Lax–Friedrichs, Roe, and FAST3 with 250 mesh points LF-type, FAST1, FAST2, and FAST3 schemes can be taken by 3/4 Again, the CFL number for the Roe-type scheme is just 1/4, since larger values could cause oscillations.The errors in the L1 norm are computed and the orders of convergence are estimated The errors, orders of convergence, the number of iterations, and the CPU time are reported by Table 1852 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Table 10 Test – Errors, orders of convergence, numbers of iterations, and CPU time of the Lax–Friedrichs-type, FAST1, FAST2, Roe-type, and FAST3 schemes LF-type N L1 -Error Order Iterations CPU time 250 500 1000 0.017835 2000 4000 0.038127 0.026825 0.59 0.011899 0.0078074 – 0.51 194 0.58 0.61 48 97 22.012 389 779 3.9468 7.8 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.027818 0.018117 0.012162 0.0079609 0.00515 – 0.62 0.57 0.61 0.63 49 97 195 390 780 4.7424 13.369 44.008 164.08 697.93 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.027034 0.017578 0.011788 0.0077084 0.004985 – 0.62 0.58 0.61 0.63 49 97 195 390 780 4.68 13.151 40.81 170.1 690.6 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.018594 0.01261 0.0080888 0.0051406 0.0033844 – 0.56 0.64 0.65 0.6 191 385 773 1550 3105 22.23 57.003 170.91 551.99 2424.9 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.018374 0.011804 0.0075361 0.0048409 0.0032385 – 0.64 0.65 0.64 0.58 54 107 214 428 857 7.2072 17.223 52.401 185.69 745.7 91.651 368.43 FAST1 FAST2 Roe-type FAST3 Fig shows the plots of the approximate solutions given by the LF-type, Roe-type, and FAST3 schemes Fig displays the approximate solutions by the FAST3 scheme with different mesh sizes As shown by Table 7, the accuracy of the Roe-type scheme is better than using the LF-type, FAST1, and FAST2 schemes However, the CPU time of the Roe-type scheme is larger than using the other ones Again, the FAST3 scheme gives the best results, where it provides smallest errors with a comparable small CPU time 4.4 Numerical test 4: solid contact in supersonic region with negative velocity In this test, we consider the Riemann problem for (1.1) and (1.2) with the initial data given in Table The exact Riemann solution begins with a 1-shock wave from U L to U This shock is 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-shock wave from U to U R These states are given in Table The solid contact between U and U are located in the supersonic region G3 and has a negative velocity us ẳ 0:1ị The structure of the Riemann solution is shown in Fig (upper-left corner) The approximate solution will be computed at the time t ¼ 0:1 on the interval ½À1; 1Š of the x-space using each of the LFtype, FAST1, FAST2and FAST3 schemes, and the Roe-type scheme (3.15) with different mesh sizes The CFL number for the LFtype, FAST1, and FAST2 can be taken by 0.9, and the CFL number for FAST3 can be taken by 0.82 Again, the CFL for the Roetype scheme is just 1/4, since larger values could cause numerical oscillations.The errors in the L1 norm are computed and the orders of convergence are estimated The errors, orders of convergence, the number of iterations, and the CPU time are reported by Table Fig shows the plots of the approximate solutions given by the LF-type, Roe-type, and FAST3 schemes Fig displays the approximate solution by the FAST3 scheme for different mesh sizes As shown by Table 11, the accuracy of the Roe-type M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 1853 Fig Test Approximate solutionusing the underlying FAST3 numerical flux The approximate solution is computed at the time t ẳ 0:1 on the interval ẵ1; with 250, 1000, and 4000 mesh points, and is compared with the exact solution Table 11 Test – The initial values Quantitiesn States UL UR ag 0.5 0.36323378 À0.62097863 18.624972 0.22134433 0.505 0.36422989 À0.97080522 18.8094 À0.42173198 pg ug ps us 1854 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Table 12 Test – The states that separate the elementary waves of the exact Riemann solution n U1 U2 U3 U4 ag 0.5 0.36323378 À0.62097863 21.413051 À0.1 0.5 0.42092135 À0.8 21.413051 À0.1 0.505 0.42207567 À0.79171392 21.625087 À0.1 0.505 0.36422989 À0.97080522 21.625087 À0.1 pg ug ps us Fig Test Approximate solution by the two-stage method using different underlying numerical fluxes: Lax–Friedrichs, Roe, and FAST3 with 250 mesh points M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 1855 Fig 10 Test Approximate solutionusing the underlying FAST3 numerical flux The approximate solution is computed at the time t ¼ 0:1 on the interval ½À1; 1Š with 250, 1000, and 4000 mesh points, and is compared with the exact solution scheme is very good However, the CPU time of the Roe-type scheme is larger than using the other ones Again, the FAST3 scheme gives the best results, where it provides better errors with a smaller CPU time than the Roe-type scheme Observe that the numerical fluxes of the FORCE and GFORCE schemes formed by convex combinations of the ones of the Lax–Friedrichs and Lax–Wendroff schemes not work for this test Thus, this failure may be caused by the computation of the Jacobian matrix and the Roe matrix of the system 4.5 Numerical test 5: mixture of water and dioxygen Consider the model (1.1), where the fluid in the g-phase is the dioxygen O2 and the fluid in the s-phase is the water H2O Both phases are considered at 100 °C, where the parameters are given by jg ¼ 4; js ¼ 5; cg ¼ 1:399; cs ¼ 1:324; 1856 M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 Table 13 Test – Errors, orders of convergence, numbers of iterations, and CPU time of the Lax-Friedrichs-type, FAST1, FAST2, Roe-type, and FAST3 schemes LF-type N L1 -Error Order Iterations CPU time 250 500 1000 2000 4000 0.31568 0.2125 0.14003 0.090155 0.056024 – 0.571 0.60173 0.63526 0.68636 80 159 318 635 1269 8.7673 16.848 46.956 148.54 591.13 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.21301 0.14108 0.090583 0.056067 0.033326 – 0.59441 0.6392 0.69209 0.7505 80 159 318 635 1269 9.4069 26.739 79.607 260.37 1060.5 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.19497 0.12852 0.081844 0.050155 0.029539 – 0.60126 0.65104 0.70648 0.76377 80 159 318 635 1269 9.7501 27.144 79.498 262.19 1039 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.71849 0.50062 0.34452 0.23394 0.15604 – 0.52125 0.53913 0.55845 0.58422 431 862 1724 3448 6895 51.075 129.37 383.25 1379.3 5519.5 N Error Order Iterations CPU time 250 500 1000 2000 4000 0.142 0.091804 0.056528 0.033406 0.01925 – 0.62926 0.69959 0.75886 0.79525 80 159 318 635 1269 9.0169 27.253 78.079 254.42 1039 FAST1 FAST2 Roe-type FAST3 see [41], for example The initial data are given in Table 11 The exact Riemann solution begins with a 3-shock from U L to U , followed by a 1-shock wave from U to U , then by a solid contact from U to U , a 2-shock from U to U , and finally it attains U R by a 4-shock wave from U These states are given in Table 12 The states U ; U on both sides of the solid contact are located in the subsonic region, where the velocity of the solid contact is negative (us ¼ À0:1) The structure of the Riemann solution is given by Fig (upper-left corner) The approximate solution will be computed at the time t ¼ 0:1 on the interval ½À1; 1Š of the x-space using each of the LFtype, FAST1, FAST2, FAST3, and the Roe-type schemes (3.15) with different mesh sizes The CFL number for the LF-type, FAST1,FAST2, FAST3 schemes can be taken by 3/4 The CFL for the Roe-type scheme is just 1/10, since larger values could cause numerical oscillations.The errors in the L1 norm are computed and the orders of convergence are estimated The errors, orders of convergence, the number of iterations, and the CPU time are reported by Table 13 All these schemes give good approximations Fig shows the plots of the approximate solutions given by the LF-type, Roe-type, and FAST3 schemes The FAST3 scheme gives the best results Fig 10 displays the approximate solution by the FAST3 scheme for different mesh sizes Conclusions and discussions In this work, a set of well-balanced schemes for the model of two-phase flows (1.1) is constructed using a numerical twostage method The set depends on a parameter h ½0; 1Š The fact that the values h ¼ and h ¼ CFL work well implies that any value h ½0; CFLŠ could work well Although the Lax–Friedrichs-type scheme has less accuracy than the Roe-type scheme, the FAST3 scheme defined by taking h ¼ CFL in this set is faster and has better accuracy than the Roe-type scheme So, schemes in this set where the parameter h takes the values near CFL could have the same property M.D Thanh / Commun Nonlinear Sci Numer Simulat 19 (2014) 1836–1858 1857 Observe that both Lax–Friedrichs and Richtmyer’s schemes not require the computations of the Jacobian matrix or the Roe matrix We note that the idea of using convex combinations of the numerical fluxes of the Lax–Friedrichs and Lax– Wendroff schemes was proposed earlier in the literature In particular, this way defines the FORCE and GFORCE schemes for the choice h ¼ 1=2 and h ẳ 1=1 ỵ CFLị, respectively However, the two-stage numerical method using the numerical fluxes of the FORCE and GFORCE schemes does not work for the model (1.1) Besides, the two-stage method using the numerical flux of the Roe scheme works only for small values of CFL numbers Therefore, this reveals that the problem could probably occur in computing the Jacobian matrix and the Roe matrix Thus, the significant feature of our method that leads to the difference in the outcomes with the FORCE and GFORCE schemes is that it does not require the computations of the Jacobian matrix or Roe matrix (See Table 10) The current work pays attention to only the isentropic case, leaving an open question for future developments for the non-isentropic case Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) 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 The Riemann problem for the Baer–Nunziato model of two-phase flows J Comput Phys 2004;195:434–64 [3] 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 [4] Baer MR, Nunziato JW A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials Int J Multi-phase Flow 1986;12:861–89 [5] Botchorishvili R, Perthame B, Vasseur A Equilibrium schemes for scalar conservation laws with stiff sources Math Comput 2003;72:131–57 [6] 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 [7] Bzil JB, Menikoff R, Son SF, Kapila AK, Steward DS Two-phase modelling of a deflagration-to-detonation transition in granular materials: a critical examination of modelling issues Phys Fluids 1999;11:378–402 [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 Vol 2004;1(4) [9] Castro MJ, Pardo A, Parés C, Toro EF On some fast well-balanced first order solvers for nonconservative systems Math Comput 2010;79:1427–72 [10] Dal Maso G, LeFloch PG, Murat F Definition and weak stability of nonconservative products J Math Pures Appl 1995;74:483–548 [11] Embid P, Baer M Mathematical analysis of a two-phase continuum mixture theory Continuum Mech Thermodyn 1992;4:279–312 [12] Godlewski E, Raviart P-A Numerical Approximation of Hyperbolic Systems of Conservation Laws Springer; 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2010 Yeom GS, Chang KS Flux-based wave decomposition scheme for an isentropic hyperbolic two-fluid model Numer Heat Transfer, Part A 2011;59:288–318  ... isentropic case may cause a certain limitation for applications, and would motivate for future developments for the general case We note that numerical approximations of nonconservative systems have attracted... Kröner D, Nam NT Numerical approximation for a Baer–Nunziato model of two-phase flows Appl Numer Math 2011;61:702–21 M.D Thanh, A well-balanced Roe-type numerical scheme for a model of two-phase flows,... materials: a critical examination of modelling issues Phys Fluids 1999;11:378–402 [8] Chinnayya A, LeRoux A- Y, Seguin N A well-balanced numerical scheme for the approximation of the shallow water