DSpace at VNU: A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section

Applied Mathematics and Computation 256 (2015) 602–629 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section Dao Huy Cuong a,b, Mai Duc Thanh c,⇑ a Nguyen Huu Cau High School, 07 Nguyen Anh Thu, Trung Chanh Ward, Hoc Mon District, Ho Chi Minh City, Viet Nam Department of Mathematics and Computer Science, University of Science, Vietnam National University-Ho Chi Minh City, 227 Nguyen Van Cu str., District 5, Ho Chi Minh City, Viet Nam c Department of Mathematics, International University, Vietnam National University-Ho Chi Minh City, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam b a r t i c l e i n f o Keywords: Fluid in a nozzle Cross-section Hyperbolic conservation law Nonconservative Riemann solver Godunov-type scheme a b s t r a c t We present a Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section The model is of nonconservative form, making it hard for standard numerical discretizations of the nonconservative term In particular, the error for a standard numerical scheme with a usual discretization of the nonconservative term may become larger as the mesh size gets smaller We first re-investigate the Riemann problem of the model, pointing out several interesting properties of the wave curves, and establishing specific existence domain for each type of solutions Then, we incorporate local Riemann solutions to build a Godunov-type scheme for the model The scheme is constructed in subsonic and supersonic regions, where the system is strictly hyperbolic Tests show that our scheme can capture standing waves, so that it is well-balanced Furthermore, tests also show that our Godunov-type scheme can give a good accuracy for numerical approximations of exact solutions Our Godunov-type scheme can resolve the difficulty of other existing schemes for similar models of fluid flows with nonconservative terms Ó 2015 Elsevier Inc All rights reserved Introduction In this paper we study to build a Godunov-type scheme for the numerical approximation of weak solutions to the initialvalue problem associated with the following isentropic model of a fluid flow in a nozzle with variable cross-section @ t aqị ỵ @ x aquị ẳ 0; @ t aquị ỵ @ x aqu2 ỵ pịị ¼ p@ x a; x R; t > 0; ð1:1Þ where qðx; tÞ; uðx; tÞ; pðx; tÞ denote the density, particle velocity, and pressure of the fluid, respectively, and a ẳ axị denotes the cross-section of the nozzle The two equations in (1.1) represent the balance of mass and momentum, respectively Even for a smooth cross-section a ẳ axị, numerical discretizations will yield piece-wise constant approximate functions Therefore, the term p@ x a is a nonconservative term, and so the system (1.1) is of nonconservative form, see [9] Numerical approximations for systems of balance laws with nonconservative terms have been a very interesting, but rather challenging topic for many authors This is because the standard numerical schemes for systems of conservation laws with usual ⇑ Corresponding author E-mail addresses: cuongnhc82@gmail.com (D.H Cuong), mdthanh@hcmiu.edu.vn (M.D Thanh) http://dx.doi.org/10.1016/j.amc.2015.01.024 0096-3003/Ó 2015 Elsevier Inc All rights reserved D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 603 discretizations of the nonconservative terms often lead to unsatisfactory results In particular, the error may become larger as the mesh sizes get smaller Furthermore, oscillations can be seen for this kind of numerical discretizations of the system Motivated by the work of LeFloch-Thanh [22] on a Godunov-type scheme for the shallow water equations with variable topography, we aim to build in this paper a Godunov-type scheme for the model of an isentropic fluid in a nozzle with variable cross-section (1.1) The most interesting result of this paper is that our Godunov-type scheme can resolve the difficulty of the well-balanced scheme in [19] when dealing with the resonant cases Moreover, the scheme can provide better convergence results than the ones in [22] for shallow water equations with variable topography Observe that a Godunov-type scheme is based on solutions of the local Riemann problem for (1.1), where, as is well-known, we supplement the system (1.1) the trivial equation @ t a ẳ 0: 1:2ị Note that the system (1.1) and (1.2) can be written in the form of a nonconservative system of balance laws @ t U ỵ AUị@ x U ẳ 0; for U ẳ q; u; aị, for example, and AUị is a matrix determined below In this work, our review on the Riemann problem for the system (1.1) and (1.2) will give us interesting properties on the wave curves together with characterizations of the existence domain of each kind of Riemann solutions when the initial data belong to different subsonic or supersonic regions Then, we employ these Riemann solutions to build up a Godunov-type scheme for the model Tests show that our Godunov-type scheme is well-balanced as it can capture steady solutions Furthermore, tests also indicate that this Godunov-type scheme possesses a good accuracy This paper continues the study on nonconservative systems of balance laws we have pursued for many years, see, for example, [20,21,28,29] for the very related topic on the Riemann problem, and [19,22,30–32] for numerical approximations The reader is referred to [23,20] for the Riemann problem for the isentropic model (1.1), to [29] for the Riemann problem for the model of a general fluid in a nozzle with discontinuous cross-section, to [21,22,7,25] for the Riemann problem for the shallow water equations with discontinuous topography, to [28,26] for the Riemann problem for two-phase flow models, and to [16,17,12] for the Riemann problem for other hyperbolic nonconservative models See [10] for the standard Godunov scheme of systems of conservation laws We note that Godunov-type schemes for various hyperbolic systems of balance laws in nonconservative form were studied in [17,8,22,2,27,26] Numerical schemes for multi-phase flow models were presented in [1,15,24,31–33] Well-balanced schemes for a single conservation law with a source term were studied in [14,5,6,11,3] Numerical schemes for other hyperbolic models in nonconservative form were presented in [4,19,18,13,30] See also the references therein The organization of this paper is as follows In Section we discuss basic concepts and properties of the system (1.1) and (1.2) Section is devoted to the revisited Riemann problem In Section we build a Godunov-type scheme for the model (1.1) Section is devoted to numerical tests Finally, we provide in Section several conclusions and discussions Preliminaries 2.1 Nonstrict hyperbolicity For simplicity, in the following we assume that the pressure is given by an equation of state of an isentropic ideal gas p ẳ pqị ẳ jqc ; where j > 0; < c < 5=3 are constant Set jc cÀ1 hqị ẳ q : c1 Observe that the function h satises h qị ẳ p0 qị=q: The system (1.1) and (1.2) for any smooth solution U ¼ ðq; u; aÞT can be re-written in the nonconservative form as @ t U ỵ AUị@ x U ẳ 0; 2:1ị where u B AUị ẳ @ h qị q qu=a u 0 C A The matrix AUị admits the following three eigenvalues k1 Uị ẳ u p p0 qị; k2 Uị ẳ u ỵ p p0 qị; k3 Uị ẳ 0: 2:2ị 604 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 The corresponding eigenvectors can be chosen as q p B C r Uị ẳ @ À p0 ðqÞ A; q p ffiffiffiffiffiffiffiffiffiffiffi B C r Uị ẳ @ p0 qị A; 0 B r Uị ẳ @ qu p0 ðqÞ Àaðu À p0 ðqÞ Þ u C A: It is evident that the first and the third characteristic speeds coincide: k1 Uị ẳ k3 Uị; on the upper sonic surface: n po Cỵ ẳ q; u; aịj u ¼ p0 ðqÞ ; and the second and the third characteristic speeds coincide: k2 Uị ẳ k3 Uị; on the lower sonic surface: n po C ẳ q; u; aịj u ẳ p0 qị ; Let ẩ ẫ C ẳ Cỵ [ C ẳ q; uịj u2 ẳ p0 ðqÞ ; ð2:3Þ which are referred to as the sonic surface Set also the regions in which the system is strictly hyperbolic: G1 ẳ fU : k1 Uị > 0g ¼ fu > pffiffiffiffiffiffiffiffiffiffiffi p0 ðqÞg; G2 ¼ fU : k1 Uị < < k2 Uịg ẳ fjuj < p p0 qịg; Gỵ2 ẳ fU G2 : u > 0g; 2:4ị G2 ẳ fU G2 : u < 0g; p G3 ẳ fU : k2 Uị < 0g ẳ fu < p0 qịg: The supersonic region is the one for which juj > pffiffiffiffiffiffiffiffiffiffiffi p0 ðqÞ; which is G1 [ G3 The subsonic region is the one for which juj < pffiffiffiffiffiffiffiffiffiffiffi p0 ðqÞ; which is G2 The third characteristic field is linearly degenerate, i.e., rk3 Uị r Uị ẳ 0; for all U in the phase domain Furthermore, it holds that c ỵ 1ị rk1 r1 ẳ rk2 r ẳ p qp00 qị ỵ 2p0 qịị ẳ 2 p0 ðqÞ pffiffiffiffiffiffi jc cÀ1 q > 0: Thus, the first and the second characteristic fields are genuinely nonlinear, i.e., rki ðUÞ Á r i ðUÞ – i ¼ 1; 2; for all U in the phase domain 2.2 Shock waves and the curves of admissible shock waves Recall that a shock wave of (1.1) and (1.2) between the two states U l ¼ ðql ; ul ; al ị; U r ẳ qr ; ur ; ar ị (al ẳ ar ) is a week solution of the form ( Ux; tị ẳ Ul ; Ur ; x < kt;  x > kt; ð2:5Þ where U l ; U r are constants, and are called the left-hand and right-hand states of the shock, respectively, and  k is a constant, which is called the shock speed Geometrically, U l is located on the left and U r is located on the right of the half line of discontinuity x ¼  kt in the ðx; tÞ-plane, see Fig D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 605 Fig A shock wave in the ðx; tÞ-plane One simple way to determine the jump relations for shocks of (1.1) and (1.2) was proposed in [20] Briefly, since the shock satisfy the conservative Eq (1.2), the usual Rankine–Hugoniot relation Àk½aŠ ¼ 0; k is the shock speed, ½aŠ ¼ ar À al is the jump of a across the discontinuity As a consequence, either the must holds, where  k vanishes, or the cross-section a remains constant across a shock shock speed  Consider the case when the cross-section a remains constant across a shock Then, the system (1.1) and (1.2) is reduced to @ t qị ỵ @ x quị ẳ 0; @ t quị ỵ @ x qu2 ỵ pị ẳ 0: The RankineHugoniot relations for the last system read kẵq ỵ ẵqu ẳ 0; kẵqu ỵ ẵqu2 ỵ p ẳ 0; 2:6ị It follows from (2.6) that given a left-hand state U ¼ ðq0 ; u0 ; a0 Þ, the Hugoniot set HðU Þ consisting of all right-hand states U ẳ q; u; a0 ị that can be connected to U by a shock is given by   HðU Þ : u ¼ u0 Ỉ j À q0  1=2 ðqc À qc0 Þ : q That i-Hugoniot curves Hi ðU Þ is tangent to eigenvector r i ðU ị; i ẳ 1; 2, at U yields  1=2 c À qc Þ À q ; ð q0 q    1=2 : H2 U ị : u ẳ u0 ỵ j q1 À q1 ðqc À qc0 Þ H1 ðU Þ : u ¼ u0 À   j Next, let us consider the admissibility criterion for shock waves As usual, we will require that any admissible shock wave of the system (1.1) and (1.2) connecting a left-hand state U l to a right-hand state U r in the genuinely nonlinear characteristic fields satisfy the Lax shock inequalities ki ðU r Þ < ki ðU l ; U r Þ < ki ðU l Þ; i ¼ 1; 2; 2:7ị ki is the shock speed, i ẳ 1; A Lax shock is a shock which satisfies the Lax shock inequalities (2.7) where  We now define the forward curves S i ðU Þ of admissible i-shock waves which consist of all right-hand states U that can be connected to a given left-hand state U by an i-Lax shock wave, i ¼ 1; It is not difficult to check that these curves are given by S U ị : u ẳ u0 S U ị : u ẳ u0 and   j q0   j 1 q0 À À q q  1=2 ðqc À qc0 Þ ; q P q0 ;  1=2 ðqc À qc0 Þ ; q q0 : The condition q P q0 for S ðU Þ (respectively q q0 for S ðU Þ) is derived from the Lax shock inequalities (2.7) Similarly, the backward i-shock wave curves S Bi ðU Þ consisting of all left-hand states U that can be connected to a given right-hand state U by an i-Lax shock wave, i ¼ 1; 2, are given by S B1 ðU ị : u ẳ u0 ỵ   j q0 À q  1=2 ðqc À qc0 Þ ; q q0 ; 606 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 and S B2 U ị : u ẳ u0 ỵ   j q0 À q  1=2 ðqc À qc0 Þ ; q P q0 : 2.3 The curves of rarefaction waves Let us now consider rarefaction waves of the system (1.1) and (1.2), which are continuous piecewise-smooth self-similar solution of the form Ux; tị ẳ Vnị; x n¼ ; t t > 0; x R: Substituting Ux; tị ẳ Vnị into the system (2.1) yields AVnịị nIịV nị ẳ 0; x nẳ : t Therefore, the integral curves of the system (1.1) and (1.2) can be determined as usual That is, the integral curves are determined by the differential equations V ðnÞ ¼ r i ðVðnÞÞ rki ðVðnÞÞ Á r i ðVðnÞÞ ; i ẳ 1; 2: 2:8ị The function > < UÀl ; Á x < ki ðU l Þt; Ux; tị ẳ V xt ; ki U l ịt x ki ðU r Þt; > : U r ; x > ki ðU r Þt; ð2:9Þ is a weak solution of (1.1) and (1.2), called an i-rarefaction wave connecting the left-hand state U l to the right-hand state U r ; i ¼ 1; We note that U l is located on the left and U r is located on the right of the rarefaction fan, see Fig Note that as a consequence of the formulas of the eigenvectors r and r2 , we have da ¼ 0; dn along the integral curves This means that the cross-section a also remain constant through any rarefaction fan Precisely, Vnị ẳ qnị; unị; aị is determined by  c1 c1 c1 qnị ẳ ql þ ðÀ1Þi pffiffiffiffi jcðcþ1Þ ðn À ki ðU l ÞÞ ; 2:10ị unị ẳ ul ỵ cỵ1 n ki U l ịị; a ẳ al ẳ ar ; ki ðU l Þ xt ki ðU r Þ: It follows from (2.8) that the forward i-rarefaction wave curves Ri ðU Þ consisting of all right-hand states U that can be connected to a given left-hand state U by an i-rarefaction wave, i ¼ 1; 2, are given by R1 U ị : u ẳ u0 R2 U ị : u ẳ u0 þ pffiffiffiffiffiffi jc ðcÀ1Þ=2 ðcÀ1Þ=2 ðq À q0 Þ; q q0 ; cÀ1 and pffiffiffiffiffiffi jc ðcÀ1Þ=2 ðcÀ1Þ=2 ðq À q0 Þ; q P q0 : cÀ1 Fig A rarefaction wave in the ðx; tÞ-plane D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 607 The backward i-rarefaction wave curve RBi ðU Þ consisting of all left-hand states U that can be connected to a given right-hand state U by an i-rarefaction wave, i ¼ 1; 2, are given by RB1 U ị : u ẳ u0 p jc ðcÀ1Þ=2 ðcÀ1Þ=2 ðq À q0 Þ; cÀ1 q P q0 ; RB2 U ị : u ẳ u0 ỵ p jc c1ị=2 c1ị=2 q q0 Þ; cÀ1 q q0 : and From the above analysis, we can not define the wave curves W U ị ẳ S U ị [ R1 U ị; W B1 U ị ẳ S B1 ðU Þ [ RB1 ðU Þ; 2:11ị W U ị ẳ S U Þ [ R2 ðU Þ; W B2 ðU ị ẳ S B2 U ị [ RB2 ðU Þ: The above argument shows that the curves W i ðU Þ; W Bi ðU Þ can be parameterized as W ðU Þ : u ẳ w1 U ; qị W B1 U ị : u ẳ wB1 U ; qị W U ị : u ẳ w2 U ; qÞ W B2 ðU Þ : u ¼ wB2 ðU ; qÞ  pffiffiffiffi  ðcÀ1Þ=2 jc > < u0 À cÀ1 qðcÀ1Þ=2 À q0 ¼  1=2 > : u0 À jð À 1Þðqc À qc Þ q0 q  1=2 > < u0 ỵ j 1ịqc qc0 ị q0 q ẳ  p  > : u0 À jc qðcÀ1Þ=2 À qðcÀ1Þ=2 cÀ1  1=2 > < u0 À jð À 1Þðqc qc0 ị q0 q ẳ  p  > : u0 ỵ jc qc1ị=2 qc1ị=2 c1  pffiffiffiffi  ðcÀ1Þ=2 jc > < u0 ỵ c1 qc1ị=2 q0 ẳ  1=2 > : u0 ỵ j 1ịqc qc ị q q ; q q0 ; ; q P q0 ; ; q q0 ; ; q P q0 ; ð2:12Þ ; q q0 ; ; q P q0 ; ; q q0 ; ; q P q0 : Accordingly, for each U ¼ ðq; uÞ, we define U2 ðU R ; UÞ as follows: U2 U R ; Uị ẳ u wB2 U R ; qÞ; wB2 ðU R ; ð2:13Þ W B2 ðU R Þ where the function qÞ is defined as (2.12) Obviously, U2 U R ; Uị ẳ for U Moreover, U2 ðU R ; UÞ > for U is above W B2 ðU R Þ and U2 ðU R ; UÞ < for U is below W B2 ðU R Þ Besides, it is not difficult to check that the wave curve W ðU ị : q # u ẳ uqị; q > is strictly decreasing, and the wave curve W B2 ðU Þ : q # uðqÞ; q > is strictly increasing 2.4 Stationary waves k must be zero The shock wave can then be obtained as the As seen above, if ½aŠ – across a shock, then the shock speed  limit of smooth stationary (i.e., time-independent) solutions This leads to the jump relations across the shock as ẵaqu ẳ 0; h i u2 ỵ hqị ẳ 0; ð2:14Þ (see [20] for the details) In this case, the shock wave connecting two states U l ; U r is called the stationary wave U l ; U r are called two equilibrium states Let U ¼ q; u; aị and U ẳ q0 ; u0 ; a0 Þ be the two equilibrium states on both sides of a stationary wave satisfying (2.14) Set l¼ 2jc : cÀ1 Assuming that U and a are fixed It follows from (2.14) that the curve W ðU Þ of stationary waves consisting of all the states U that can be connected to U by a stationary wave can be parameterized by q Precisely, the curve W ðU Þ is given by W U ị :   c1 1=2 u ẳ w3 U ; qị :ẳ sgnu0 ị u20 lqc1 q0 ị ; a ẳ aqị ẳ qwa03qU0 u0 0;qÞ : ð2:15Þ 608 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Eliminating u ¼ w3 ðU ; qÞ in (2.15), we obtain the following nonlinear equation in q  1=2 a u q Fqị ẳ sgnu0 ị u20 lqc1 qc01 ị q 0 ẳ 0: a 2:16ị The domain of the function F is given by  ðU ị ẳ 06q6q  l u20 ỵ qc01 cÀ1 : We have cÀ1 @FðqÞ u20 À lðqcÀ1 q0 ị jcqc1 ẳ  :  @q cÀ1 1=2 u20 À lðqcÀ1 À q0 Þ Consider first u0 > The last expression means that @FðqÞ @q > 0; q < qmax ðU Þ; @FðqÞ @q < 0; q > qmax ðU Þ; where qmax U ị ẳ   c1 u20 þ lq0cÀ1 : lðc þ 1Þ ð2:17Þ The function q # FðqÞ takes negative values at the end-points 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 2:18ị For u0 < 0, similar properties hold Thus, given U , a stationary wave connecting from U to some state U ẳ q; u; aị exists if and only if a P amin ðU Þ Moreover, if a > amin ðU Þ, then there are exactly two values u1 ðU ; aÞ < qmax ðU Þ < u2 ðU ; aÞ such that Fðu1 ðU ; aịị ẳ Fu2 U ; aịị ẳ 0: ð2:19Þ The following lemma characterizes the stationary waves Lemma 2.1 [20, Lem 2.3] The following conclusions hold (a) qmax ðU Þ > q0 ; U G1 [ G3 ; qmax ðU Þ < q0 ; U G2 ; qmax U ị ẳ q0 ; U Cặ : 2:20ị (b) The state ðu1 ðU ; aÞ; w3 ðU ; u1 ðU ; aÞÞ from the other side of a stationary jump from U belongs to G1 if u0 > 0, and belongs to G3 if u0 < 0, while the state ðu2 ðU ; aÞ; w3 ðU ; u2 ðU ; aÞÞ belongs to G2 In addition, it holds that (i) If a > a0 , then u1 ðU ; aÞ < q0 < u2 ðU ; aÞ: ð2:21Þ (ii) If a < a0 , then q0 < u1 ðU ; aÞ for U G1 [ G3 ; q0 > u2 ðU ; aÞ for U G2 : ð2:22Þ (c) a > amin ðUÞ; a ¼ amin ðU; aÞ; ðq; u; aÞ Gi ; i ẳ 1; 2; 3; q; u; aị C Æ : ð2:23Þ It follows from Lemma 2.1 that there are two possible stationary waves from a given state U to a state with a new level cross-section a Thus, it is necessary to impose some condition to select a unique physical stationary wave as follows D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 609 (MC) Along the stationary wave curve W ðU Þ defined by (2.15), the component a has to be monotone with respect to q Furthermore, the total variation of the cross-section component of any Riemann solution must not exceed jaR À aL j, where aR ; aL are the cross-sections at the left-hand and right-hand states, respectively Observe that the admissibility criterion (MC) implies that any stationary jump must not cross the sonic curve in the ðq; uÞplane k1 ;  k2 may vanish, and therefore can coincide with the zero-shock speed of the It is interesting to see that the shock speeds  stationary shocks, as indicated in the following lemma Lemma 2.2 [20, Prop 2.4] k1 ðU ; UÞ may change sign along the forward 1-shock curve S ðU Þ More precisely, if U G2 [ G3 then (i) The 1-shock speed   k1 ðU ; UÞ remains negative: k1 ðU ; UÞ < 0; U S ðU Þ: If U G1 , then there is exactly one state, denoted by ð2:24Þ U# S U ị \ Gỵ 2, k1 U ; U # ị ẳ 0; k1 ðU ; UÞ > 0; q < q < q# ; 0 k1 ðU ; UÞ < 0; q > q# : such that ð2:25Þ k2 ðU ; UÞ may change sign along the backward 2-shock curve S B2 ðU Þ More precisely, if U G1 [ G2 then (ii) The 2-shock speed   k2 ðU ; UÞ remains positive: k2 ðU ; UÞ > 0; U S B2 ðU Þ: ð2:26Þ À If U G3 , then there is exactly one state U # S ðU Þ \ G2 , such that k2 ðU ; U # Þ ¼ 0; k2 ðU ; UÞ < 0; q0 < q < q# 0; k2 ðU ; UÞ > 0; q > q# : ð2:27Þ The Riemann problem revisited In this section the Riemann problem for (1.1) and (1.2) is revisited, (see also [20]) Solutions of the Riemann problem, or Riemann solutions, are made of a finite number of elementary waves, which are Lax shocks, rarefaction waves, or admissible stationary waves It is sufficient to consider only the Riemann data in G1 [ Cỵ [ G2 ; aR > aL , since the other cases can similarly be obtained The constructions will be based on the left-hand state U L , and we distinguish between two cases:  Case A: U L G1 [ Cỵ ;  Case B: U L G2 Notations (i) W k ðU i ; U j Þ(Sk ðU i ; U j Þ; Rk ðU i ; U j Þ) denotes the kth-wave (kth-shock, kth-rarefaction wave, respectively) connecting the left-hand state U i to the right-hand state U j ; k ¼ 1; 2; 3; (ii) W m ðU i ; U j Þ È W n ðU j ; U k Þ indicates that there is an mth-wave from the left-hand state U i to the right-hand state U j , followed by an nth-wave from the left-hand state U j to the right-hand state U k ; m; n f1; 2; 3g (iii) U # denotes the state resulting from a zero-speed shock wave from U (iv) U denotes the state resulting from a stationary wave from U 3.1 Case A: U L G1 [ Cỵ In this subsection we construct three composite wave curves W 31 ðU L Þ; W 313 ðU L Þ and W 13 ðU L Þ corresponding to U L G1 [ Cỵ in order to build three constructions A1, A2 and A3 Then, it is interesting that we can point out a ‘‘large enough’’ neighborhood containing U L such that the Riemann problem for (1.1) and (1.2) admits a solution whenever U R belongs this neighborhood 610 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Construction A1 We construct the composite wave curve W 31 ðU L Þ as follows First, the solution begins with a stationary wave from U L to U ¼ U 0L using smaller root u1 ðU L ; aR Þ to shift aL to aR , where U 0L ¼ ðq0L ; u0L ; aR Þ W ðU L Þ; ð3:1Þ q0L ¼ u1 ðU L ; aR Þ: At the state U 0L , we have the characteristic speed k1 ðU 0L Þ > It is known that there is only one state U 0# such that L U 0# L S ðU L Þ; k1 ðU 0L ; U 0# L ị ẳ 0; 3:2ị k1 ðU 0L ; UÞ > 0; q0L < q < q0# L ; k1 ðU L ; UÞ < 0; q > q0# : L Second, the next part of the solution is a 1-wave from U ¼ U 0L to a state U W ðU 0L Þ such that q q0# L Last, the composite wave curve W 31 ðU L Þ is defined as W 31 ðU L Þ ¼ fUðq; u; aR Þ : U W ðU 0L Þ; q q0# L g: ð3:3Þ We denote U up ẳ W 31 U L ị \ fq ẳ 0g: 3:4ị Obviously, U up and U 0# are two end-points of W 31 ðU L Þ See Fig Therefore, if the following holds L U2 ðU R ; U up Þ:U2 ðU R ; U 0# L Þ < 0; ð3:5Þ we will have an intersection U ẳ W B2 U R ị \ W 31 ðU L Þ; ð3:6Þ consequently, the Riemann problem for (1.1) and (1.2) has a solution of the form W ðU L ; U Þ È W ðU ; U Þ È W ðU ; U R Þ: ð3:7Þ Remark 3.1 The solution (3.7) always makes sense Indeed, first, if U is below U R on W B2 ðU R Þ then W ðU ; U R Þ is a 2rarefaction wave R2 ðU ; U R Þ Since U 2 W 31 ðU L Þ & G1 [ Cỵ [ Gỵ , then U R belongs to G1 [ Cỵ [ G2 ; therefore k2 ðU R Þ > and (3.7) makes sense Second, if U is above U R on W B2 ðU R Þ then W ðU ; U R Þ is a 2-shock wave S2 ðU ; U R Þ; therefore (3.7) also  makes sense, since U 2 G1 [ Cỵ [ Gỵ and k2 ðU R ; U Þ > Construction A2 We construct the composite wave curve W 313 ðU L Þ as follows First, for each cross-section level aM ½aL ; aR Š, the solution begins with a stationary wave from U L to U ¼ U M using smaller root u1 ðU L ; aM Þ to shift aL to aM , where U M ¼ ðqM ; uM ; aM Þ W ðU L ị; qM ẳ u1 U L ; aL ị: Fig The composite wave curves W 31 ðU L Þ; W 313 ðU L Þ; W 13 ðU L Þ corresponding to constructions A1, A2, A3 ð3:8Þ D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 611 Second, the next part of the solution is a zero-speed 1-shock from U ¼ U M to state U ¼ U # M , i.e U2 ¼ U# M S ðU M Þ; ð3:9Þ k1 U M ; U # M ị ẳ 0: #0 Third, the next part of the solution is a stationary wave from U ¼ U # M to state U ¼ U M using bigger root u2 ðU ; aR Þ to shift aM to aR , where U #0 M ¼ ðq3 ; u3 ; aR Þ W ðU Þ; ð3:10Þ q3 ¼ u2 ðU ; aR Þ: Last, the composite wave curve W 313 ðU L Þ is defined as W 313 U L ị ẳ fU #0 M : aL aM aR g: ð3:11Þ 0# 0# #0 It is not difficult to check that U #0 L and U L are two end-points of W 313 ðU L Þ, where U L is defined as (3.2) and U L is defined as U# L S ðU L Þ; U #0 L #0 L q k1 ðU L ; U # L ị ẳ 0; #0 #0 L ; uL ; aR Þ # ðU L ; aR ị: ẳ q ẳu W U # L Þ; ð3:12Þ See Fig Therefore, if the following holds #0 U2 ðU R ; U 0# L Þ:U2 ðU R ; U L Þ < 0; ð3:13Þ we will have an intersection U W B2 ðU R Þ \ W 313 ðU L Þ; ð3:14Þ consequently, the Riemann problem for (1.1) and (1.2) has a solution of the form W ðU L ; U Þ È S1 ðU ; U Þ È W ðU ; U Þ È W ðU ; U R Þ: ð3:15Þ Remark 3.2 The solution (3.15) always makes sense since U W 313 U L ị & Gỵ Construction A3 We construct the composite wave curve W 13 ðU L Þ as follows First, the solution begins with 1-shock wave from U L to a state U , where U S ðU L Þ; ð3:16Þ k1 ðU L ; U Þ 0: So, U is located between U # L and U À on S ðU L Þ, where U À ¼ W ðU L Þ \ CÀ : ð3:17Þ Second, the next part of the solution is a stationary wave from U to U ¼ where U 01 ẳ q01 ; u01 ; aR ị W ðU Þ; U 01 using bigger root u2 ðU ; aR Þ to shift aL to aR , 3:18ị q01 ẳ u2 U ; aR ị: Last, the composite wave curve W 13 ðU L Þ is dened as W 13 U L ị ẳ fU 01 : Obviously, U 0À À U #0 L and U 0À U ðq; u; aL Þ S ðU L Þ; q#L q qÀ g: ð3:19Þ are two end-points of this curve, where 0 ; u ; aR ị ẳ q W U ị; q ẳ u2 U ; aR Þ: ð3:20Þ See Fig Therefore, if the following holds U2 ðU R ; U #0 L Þ:U2 ðU R ; U À Þ < 0; ð3:21Þ we will have an intersection U 2 W B2 ðU R Þ \ W 13 ðU L Þ; ð3:22Þ hence, the Riemann problem for (1.1) and (1.2) has a solution of the form S1 ðU L ; U Þ È W ðU ; U Þ È W ðU ; U R Þ; provide that  k2 ðU R ; U Þ > when W ðU ; U R Þ is a 2-shock wave S2 ðU ; U R Þ ð3:23Þ D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 U ¼ ðq1 ; u1 ; aL ị W U L ị; qỵ q1 qÀ : 615 ð3:45Þ Obviously, U is located between U ỵ and U on W ULị, where U ỵ is dened as (3.24) and U À is defined as U À ¼ ðqÀ ; u ; aL ị ẳ W U L ị \ CÀ : ð3:46Þ Second, the next part of the solution is a stationary wave from U to U 01 using bigger root u2 ðU ; aR Þ to shift aL to aR , where U 01 ¼ ðq01 ; u01 ; aR Þ W ðU ị; 3:47ị q01 ẳ u2 U ; aR Þ: Last, the composite wave curve W 13 ðU L Þ is defined as W 13 ðU L Þ ¼ fU 01 : U ¼ ðq1 ; u1 ; aL ị W U L ị; qỵ q1 q g: 3:48ị Obviously, U 0ỵ and U 0À are two endpoints of the composite curve W 13 U L ị, where U 0ỵ is dened as (3.38) and U 0À is defined as (3.20) See Fig Therefore, if the following holds U2 ðU R ; U 0ỵ ị:U2 U R ; U ị < 0; ð3:49Þ we will have an intersection U 2 W B2 ðU R Þ \ W 13 ðU L Þ; ð3:50Þ hence, the Riemann problem for (1.1) and (1.2) has a solution of the form W ðU L ; U Þ È W ðU ; U Þ È W ðU ; U R Þ; ð3:51Þ k2 ðU R ; U Þ > when W ðU ; U R Þ is a 2-shock wave S2 ðU ; U R Þ provide that  Remark 3.6 Since U may belong to GÀ , then the solution (3.51) may not make sense By providing a condition k2 ðU R ; U Þ > when W ðU ; U R Þ is a 2-shock wave S2 ðU ; U R Þ and k2 ðU R ị > when U R G1 [ Cỵ [ G2 , we will have  W ðU ; U R Þ is a 2-rarefaction wave R2 ðU ; U R Þ; therefore the solution (3.51) makes sense Similar to the Theorem 3.1, we also have the following theorem Theorem 3.2 Assume that U L G2 and aR > aL Consider the open set OB and the rectangle RB (see Fig 6) defined as: n OB ¼ U à ¼ ðqà ; uà ; aR ị : ẩ RB ẳ U ẳ q ; uà ; aR Þ : o U2 ðU à ; U ỵup ị:U2 U ; U ị < ; < qà < q0À ; É u0 < u < uỵup : 3:52ị It is hold that:  RB is a subset of OB ;  if U R belongs to RB \ G1 [ Cỵ [ G2 Þ, the Riemann problem for (1.1) and (1.2) has a solution of the form (3.35), or of the form (3.44), or of the form (3.51) Building a Godunov-type numerical scheme Relying on the constructions of Riemann solutions in the previous section, we are now in a position to build up a Godunov-type scheme As seen later, this numerical scheme has a quasi-conservative property Let us set 1 q qu qu 1B B C B C C U ẳ @ qu A; FUị ẳ @ qu2 ỵ p A; SUị ẳ @ qu2 A: a a ð4:1Þ Then, the system (1.1) and (1.2) can be written in the compact form @ t U ỵ @ x FUị ẳ SUị@ x a; t > 0; x R: ð4:2Þ Accordingly, given the initial condition Ux; 0ị ẳ U xị; x R; then, the discrete initial values U 0j ¼ Dx Z ð4:3Þ ðU 0j Þj2Z are given by xjỵ1=2 U xịdx: xj1=2 Now, suppose U n ẳ ðU nj Þj2Z is known We define the approximation U nỵ1 ẳ U nỵ1 ịj2Z of U:; t nỵ1 ị as follows: j ð4:4Þ 616 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 (i) We extend the sequence U n as a piecewise constant function U D ð:; tn Þ defined by U D ðx; t n ị ẳ U nj ; xj1=2 < x < xjỵ1=2 : 4:5ị (ii) We solve the Cauchy problem for (4.2) with the initial condition Ux; 0ị ẳ U D ðx; t n Þ; ð4:6Þ to find the solution Uð:; DtÞ This solution is obtained by solving a juxtaposition of local Riemann problems and Ux; Dtị ẳ U x x  jỵ1=2 ; U nj ; U njỵ1 ; Dt xj < x < xjỵ1 ; j Z; ð4:7Þ À Á where U xt ; U L ; U R denotes the exact solution of the Riemann problem for (4.2) corresponding to the Riemann data ðU L ; U R Þ (iii) We project (L2 -projection) the exact solution Uð:; DtÞ onto the piecewise constant functions, i.e., we set U nỵ1 ẳ j Dx Z xjỵ1=2 Ux; DtÞdx: ð4:8Þ xjÀ1=2 We also assume that the following C.F.L condition holds: Dt maxfjkk U nj ịj; Dx k ẳ 1; 2; 3g ; ð4:9Þ where kk ; k ¼ 1; 2; are the eigenvalues of the matrix AðUÞ given by (2.2), so that the waves issued from the points xj1=2 and xjỵ1=2 not interact It is not difficult to check that the a-component is constant in each interval xj1=2 ; xjỵ1=2 ị, so the right-hand side of (4.2) vanishes Thus, the scheme is in quasi-conservative form in the sense that U nỵ1 ẳ U nj À j  Dt  FðUð0À; U nj ; U njỵ1 ịị FU0ỵ; U nj1 ; U nj ÞÞ : Dx ð4:10Þ This scheme is capable of capturing stationary waves exactly Therefore, it is a well-balanced scheme To see this, we observe that if U n corresponds to a stationary wave, then on each cell xjÀ1=2 < x < xjỵ1=2 ; t n < t t nỵ1 , the exact Riemann solution is constant, j Z; n ¼ 0; 1; 2; This implies that U0; U nj ; U njỵ1 ị ẳ U0ỵ; U nj1 ; U nj ị; which yields U nỵ1 ẳ U nj ; j n ¼ 0; 1; 2; So, if the initial data U corresponds to a stationary wave, then it holds by induction that for all j Z; U nj ¼ U 0j ; for all j Z and n ¼ 0; 1; 2; The last equality means that the scheme is capable of capturing stationary waves exactly To complete the definition of the Godunov-type scheme (4.10) we need to define the values U0ặ; U L ; U R ị as follows Riemann solver (A1) The Riemann solver (A1) relying on Construction A1 yields Uð0À; U L ; U R Þ ẳ U L ; 4:11ị U0ỵ; U L ; U R ị ẳ U 0L ; where U 0L ẳ ðq0L ; u0L ; aR Þ W ðU L ị; q0L ẳ u1 U L ; aR ị: ð4:12Þ This implies that the Godunov-type scheme (4.10) using the Riemann solver (A1) becomes U nỵ1 ẳ U nj j  Dt  FðU nj Þ À FððU njÀ1 Þ Þ ; Dx ð4:13Þ where ðU njÀ1 Þ is defined as (4.12) Riemann solver (A2) We denote U A2 ðU L ; U R Þ the point satisfying U A2 U L ; U R ị ẳ W 313 ðU L Þ \ W B2 ðU R Þ: ð4:14Þ The Riemann solver (A2) relying on Construction A2 yields U0; U L ; U R ị ẳ U L ; U0ỵ; U L ; U R ị ẳ U A2 ðU L ; U R Þ: ð4:15Þ 617 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 This implies that the Godunov-type scheme (4.10) using the Riemann solver (A2) becomes U nỵ1 ẳ U nj À j  Dt  FðU nj Þ À FðU A2 ðU njÀ1 ; U nj ÞÞ ; Dx ð4:16Þ where U A2 ðU njÀ1 ; U nj Þ is defined as (4.14) We sketch an algorithm for computing U A2 ðU L ; U R Þ as follows First, we set a1 ¼ aL ; a2 ¼ aR  Step 1: Estimate aM ẳ a1 ỵa , – compute U M ¼ ðqM ; uM ; aM Þ W ðU L Þ, where qM ẳ u1 U L ; aM ị, # compute U # M S ðU M Þ such that k1 U M ; U M ị ẳ 0, # # #0 #0 #0 – compute U #0 M ẳ qM ; uM ; aR ị W U M ị, where qM ẳ u2 U M ; aR Þ  Step 2: #0 B – If U #0 M W ðU R Þ, terminate the computation and set U A2 ðU L ; U R ị ẳ U M , B if U #0 M is above the curve W ðU R Þ, set a1 ¼ aM and return step 1, B – if U #0 M is below the curve W U R ị, set a2 ẳ aM and return step Riemann solver (A3) We denote U A3 ðU L ; U R Þ and U 0A3 ðU L ; U R Þ two states satisfying: U A3 ðU L ; U R Þ W ðU L Þ from U # L U 0A3 ðU L ; U R Þ U 0A3 ðU L ; U R Þ to U À ; W ðU A3 U L ; U R ịị; ẳ W 13 U L Þ \ ð4:17Þ W B2 ðU R Þ: The Riemann solver (A3) relying on Construction A3 yields Uð0À; U L ; U R ị ẳ U A3 U L ; U R ị; 4:18ị U0ỵ; U L ; U R ị ẳ U 0A3 U L ; U R Þ: This implies that the Godunov-type scheme (4.10) using the Riemann solver (A3) becomes U nỵ1 ẳ U nj j  Dt  FðU A3 ðU nj ; U njỵ1 ịị FU 0A3 U nj1 ; U nj ÞÞ ; Dx ð4:19Þ where U A3 ðU nj ; U njỵ1 ị and U 0A3 U nj1 ; U nj Þ are defined as (4.17) We sketch an algorithm for computing two states U A3 ðU L ; U R Þ and U 0A3 ðU L ; U R Þ as follows First, set q1 ¼ q# L ; q2 ¼ qÀ  Step 1: – Estimate qT ẳ q1 ỵ2 q2 , compute U T ẳ ðqT ; uT ; aL Þ W ðU L ị, compute U 0T ẳ q0T ; u0T ; aR Þ W ðU T Þ, where q0T ẳ u2 U T ; aR ị  Step 2: – If U 0T W B2 ðU R Þ, terminate the computation and set U A3 ðU L ; U R ị ẳ U T ; U 0A3 U L ; U R ị ẳ U 0T , – if U 0T is above W B2 ðU R Þ, set q1 ¼ qT and reture step 1, – if U 0T is below W B2 ðU R Þ, set q2 ¼ qT and reture step Riemann solver (B1) The Riemann solver (B1) relying on Construction B1 yields U0; U L ; U R ị ẳ U ỵ ; 4:20ị U0ỵ; U L ; U R ị ẳ U 1ỵ ; where U ỵ ẳ qỵ ; uỵ ; aL ị ẳ W U L ị \ Cỵ ; U 1ỵ ẳ q1ỵ ; u1ỵ ; aR ị W U ỵ ị; q1ỵ ẳ u1 U ỵ ; aR ị: 4:21ị This implies that the Godunov-type scheme (4.10) using the Riemann solver (B1) becomes U nỵ1 ẳ U nj j  Dt  FU nj ịỵ ị FU nj1 ịỵ ị ; Dx where U nj ịỵ and U nj1 ịỵ are defined as (4.21) ð4:22Þ 618 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Riemann solver (B2) We denote U B2 ðU L ; U R Þ the point satisfies U B2 ðU L ; U R ị ẳ W 1313 U L ị \ W B2 ðU R Þ: ð4:23Þ It is easy to check that U B2 U L ; U R ị ẳ U A2 U ỵ ; U R ị: The Riemann solver (B2) relying on Construction B2 (3.9) yields Uð0À; U L ; U R ị ẳ U L ; U0ỵ; U L ; U R ị ẳ U B2 U L ; U R Þ: ð4:24Þ This implies that the Godunov-type scheme (4.10) using the Riemann solver (B2) becomes U nỵ1 ẳ U nj j  Dt  FU nj Þ À FðU B2 ðU njÀ1 ; U nj ÞÞ ; Dx where U B2 ðU njÀ1 ; U nj Þ is defined as (4.23) Fig Test An exact stationary wave and its approximation at the time t ¼ 0:1 on the interval [À1, 1] with 1000 mesh points ð4:25Þ D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 619 Riemann solver (B3) We denote U B3 ðU L ; U R Þ and U 0B3 ðU L ; U R Þ two states satisfying: U B3 ðU L ; U R Þ WU L ị from U ỵ to U ; U 0B3 ðU L ; U R Þ W ðU B3 ðU L ; U R ÞÞ; U 0B3 U L ; U R ị ẳ W 13 ðU L Þ \ ð4:26Þ W B2 ðU R Þ: The computing algorithm of two states U B3 ðU L ; U R Þ and U 0B3 ðU L ; U R Þ is similar to computing algorithm of U A3 ðU L ; U R Þ and U 0A3 ðU L ; U R Þ The Riemann solver (B3) relying on Construction B3 yields Uð0À; U L ; U R ị ẳ U B3 U L ; U R ị; 4:27ị U0ỵ; U L ; U R ị ẳ U 0B3 ðU L ; U R Þ: This implies that the Godunov-type scheme (4.10) using the Riemann solver (B3) becomes U nỵ1 ẳ U nj j  Dt  FU B3 U nj ; U njỵ1 ịị FðU 0B3 ðU njÀ1 ; U nj ÞÞ ; Dx 4:28ị where U B3 U nj ; U njỵ1 ị and U 0B3 ðU njÀ1 ; U nj Þ are defined as (4.26) Numerical experiments This section is devoted to numerical tests by using MATLAB, which demonstrate the efficiency of our scheme (4.10) For each test, we compare the numerical solution U h with the corresponding exact solution U By using the stability condition CFL ¼ 0:75; and taking j ¼ 1; c ¼ 1:6, we plot the solutions U h and U for x ½À1; 1Š; t ¼ 0:1: 5.1 Test Let the Riemann data be given by U L ¼ ðpL ; uL ; aL Þ ¼ ð1; 2; 1:5Þ G1 ; U R ¼ ðpR ; uR ; aR Þ ¼ ð0:326848262627227; 2:413875590220778; 2:5Þ G1 : ð5:1Þ It is easy to check that the data (5.1) satisfies the jump relation for stationary wave (2.14) Therefore, the solution is just a stationary wave from U L to U R See Fig Fig displays an exact stationary wave and its approximation at the time t ¼ 0:1 on the interval [À1, 1] with 1000 mesh points This figure shows that our Godunov-type scheme can capture stationary solutions, so it is well-balanced The errors for Test are reported in the Table Table Errors for Test N kU h À UkL1 kU h À UkL1 =kUkL1 250 0.0060 1:0105  10À3 1000 0.0030 5:0540  10À4 Table States that separate the elementary waves of the exact solution of the Riemann problem in Test Pressure Velocity Cross-section U1 U2 0.326848262627227 2.413875590220778 2.5 0.036589595247276 3.565104955362759 2.5 620 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Table Errors of numerical approximations for different mesh sizes for Test N kU h À UkL1 kU h À UkL1 =kUkL1 250 500 1000 0.087 0.055 0.032 0.012 0.008 0.004 Fig Test Exact solution and approximate solutions for different mesh sizes 621 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 5.2 Test In this test we approximate a Riemann solution of Construction A1 Precisely, we consider the Riemann data to be U L ẳ pL ; uL ; aL ị ¼ ð1; 2; 1:5Þ G1 ; ð5:2Þ U R ¼ ðpR ; uR ; aR Þ ¼ ð0:5; 5; 2:5Þ G1 : We have U2 ðU R ; U up ị ẳ 4:535219207219139 > 0; U2 U R ; U 0# L ị ẳ 5:650839239122051 < 0: Therefore, according to Construction A1, the exact solution is a stationary wave from U L to U ¼ U 0L , followed by a 1-rarefaction wave from U to U ẳ W 31 U 0L ị \ W B2 ðU R Þ, then followed by a 2-rarefaction wave from U to U R , where U ; U are reported in Table The errors for Test are reported in the Table Fig displays the exact solution at the time t ¼ 0:1 on the interval x ½À1; 1Š and its approximate solutions corresponding to the mesh sizes h ¼ 1=250 and h ¼ 1=1000 This figures shows that the approximate solutions get closer to the exact solution when the mesh sizes get smaller Table shows that the errors become smaller as the mesh size gets smaller So, Test demonstrates the convergence of the approximate solutions by the Godunov-type scheme to the exact solution 5.3 Test In this test we approximate a Riemann solution of Construction A2 Consider the initial data: U L ¼ ðpL ; uL ; aL ị ẳ 13:1; 4; 1:5ị G1 ; 5:3ị U R ẳ pR ; uR ; aR ị ẳ ð269; 4; 2:5Þ G1 : We have U2 ðU R ; U 0# L ị ẳ 0:500824057542087 > 0; U2 U R ; U #0 L ị ẳ 0:605187142658357 < 0: Therefore, according to Construction A2, the exact solution is a stationary wave from U L to U , followed by a 1-shock wave with zero-speed from U to U , followed by again a stationary wave from U to U W 313 ðU L Þ \ W B2 ðU R Þ, then followed by a 2-rarefaction wave from U to U R , where U ; U ; U are reported in Table The errors for Test are reported in the Table The exact solution at the time t ¼ 0:1 on the interval x ½À1; 1Š and its approximations for different mesh sizes h ¼ 1=250 and h ¼ 1=1000 are displayed in Fig We can see from this figure that the approximate solutions are closed to the exact solution Furthermore, Table shows that the errors become smaller as the mesh size gets smaller So, Test also indicates that the approximate solutions by the Godunov-type scheme are convergent to the exact solution when the initial data belong to the supersonic region Table States that separate the elementary waves of the exact solution of the Riemann problem in Test p u a U1 U2 U3 7.131019501417605 4.342109858071679 2.020808862711419 53.152187297315628 1.237285316741408 2.020808862711419 56.899158001686963 0.958439384702954 2.5 Table Errors of numerical approximations for different mesh sizes for Test N kU h À UkL1 kU h À UkL1 =kUkL1 250 500 1000 4.780 2.897 1.646 0.030 0.018 0.010 622 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Fig Test Exact solution and approximate solutions for different mesh sizes Table States that separate the elementary waves of the exact solution of the Riemann problem in Test p u a U1 U2 16.367849830811508 À1.562231548681802 1.5 22.193029608466986 À0.774918544249643 2.5 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 623 Fig 10 Test Exact solution and approximate solutions for different mesh sizes Table Errors of numerical approximations for different mesh sizes for Test N kU h À UkL1 kU h À UkL1 =kUkL1 250 500 1000 0.549 0.267 0.133 0.025 0.012 0.006 5.4 Test In this test we approximate a Riemann solution of Construction A3, when the initial data belong to both dies of the sonic curve Cỵ Precisely, we consider the Riemann problem for (1.1) and (1.2) with the initial data of the form 624 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Table States that separate the elementary waves of the exact solution of the Riemann problem in Test p u a Uỵ U1 U2 0.377368057812661 1.053667184251945 1.5 0.081088548662761 1.652849119001248 2.5 0.035171088474970 2.034490265394593 2.5 Fig 11 Test Exact solution and approximate solutions for different mesh sizes 625 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 U L ẳ pL ; uL ; aL ị ẳ 1; 2; 1:5ị G1 ; 5:4ị U R ẳ pR ; uR ; aR ị ẳ 13; 1:5; 2:5ị G2 : We have U2 ðU R ; U #0 L ị ẳ 3:433764548414430 > 0; U2 U R ; U ị ẳ 1:100882768413456 < 0: Therefore, according to Construction A3, the exact solution is a 1-shock wave from U L to U , followed by a stationary wave from U to U 2 W 13 ðU L Þ \ W B2 ðU R Þ, then followed by a 2-shock wave from U to U R , where U ; U are reported in Table The exact solution and its approximations at the time t ¼ 0:1 on the interval [À1, 1] corresponding to the mesh sizes h ¼ 1=250 and h ¼ 1=1000 are shown by Fig 10 This figures illustrates good approximations to the exact solution The errors for Test are reported in the Table 7, where one can see that the errors become smaller as the mesh size gets smaller So, Test demonstrates the convergence of the approximate solutions by the Godunov-type scheme to the exact solution when the initial data belong to both supersonic and subsonic regions 5.5 Test In this test we approximate a Riemann solution of Construction B1 when the initial data also belong to both supersonic and subsonic regions Precisely, let us consider the Riemann problem for (1.1) and (1.2) with the initial data of the form U L ¼ ðpL ; uL ; aL ị ẳ 9; 1:8; 1:5ị G2 ; 5:5ị U R ẳ pR ; uR ; aR ị ẳ 1; 4; 2:5ị G1 : We have U2 U R ; U ỵup ị ẳ 4:501720957903902 > 0; U2 U R ; U 1# ỵ ị ¼ À2:861867429598399 < 0: Therefore, according to Construction B1, the exact solution is a 1-rarefaction wave from U L to U ỵ , followed by a stationary wave from U þ to U , followed by a 1-rarefaction wave from U to U ¼ W 131 ðU L Þ \ W B2 ðU R Þ, then followed by a 2-rarefaction wave from U to U R , where U ỵ ; U ; U are reported in Table As shown by Test in [30], the well-balanced scheme in [19] does not provide good approximations to the exact solution in this case The same difficulty for the Godunov-type for the shallow water equations with variable topography [22], where a non-convergence case was obtained in the resonant regime However, we can see below that our Godunov-type scheme can provide us with very reliable approximations of the exact solution Fig 11 displays the exact solution at the time t ¼ 0:1 on the interval x ½À1; 1Š and its approximations for different mesh sizes h ¼ 1=250 and h ¼ 1=1000 Fig 11 shows that the approximate solutions are closed to the exact solution Furthermore, the approximate solution corresponding to a finer mesh size is closer to the exact solution than the one corresponding to a larger mesh size The errors for Test are reported in the Table 9, which indicate that the errors become smaller as the mesh size gets smaller Thus, Test demonstrates the convergence of the approximate solutions by the Godunov-type scheme to the exact solution when the initial data also belong to both supersonic and subsonic regions This suggests that our Godunovtype scheme can resolve the difficulty by the well-balanced scheme proposed by [19] Furthermore, this result is stronger than the one for the shallow water equations [22] Table Errors of numerical approximations for different mesh sizes for Test N kU h À UkL1 kU h À UkL1 =kUkL1 250 500 1000 0.256 0.152 0.089 0.021 0.013 0.007 Table 10 States that separate the elementary waves of the exact solution of the Riemann problem in Test p u a Uỵ U1 U2 U3 0.377368057812661 1.053667184251945 1.5 0.126661623744697 1.534094549577439 2.038297820021398 0.498990208841633 0.651175075314172 2.038297820021398 0.559289712949465 0.494378864453038 2.5 626 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Table 11 Errors of numerical approximations for different mesh sizes for Test N kU h À UkL1 kU h À UkL1 =kUkL1 250 500 1000 2000 0.60 0.35 0.21 0.12 0.029 0.017 0.01 0.006 Fig 12 Test Exact solution and approximate solutions for different mesh sizes D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 627 5.6 Test In this test we approximate a Riemann solution of the very complicated situation of Construction B2, where there is a shock collision: an exact solution contain three waves of the same zero shock speed Let us take the Riemann data as U L ẳ pL ; uL ; aL ị ẳ 9; 1:8; 1:5ị G2 ; 5:6ị U R ẳ pR ; uR ; aR ị ẳ 18:5; 4; 2:5ị G1 : We have U2 ðU R ; U 1# ỵ ị ẳ 0:208505075826081 > 0; U2 U R ; U 0ỵ ị ẳ 0:200866931644104 < 0: Therefore, according to Construction B2, the exact Riemann solution begins with a 1-rarefaction wave from U L to U ỵ , followed by a stationary wave from U ỵ to U , followed by a 1-shock wave with zero-speed from U to U , followed by again a stationary wave from U to U W 1313 ðU L Þ \ W B2 ðU R Þ, then followed by a 2-rarefaction wave from U to U R , where U ỵ ; U ; U ; U are reported in Table 10 The errors for Test are reported in the Table 11 The errors, as seen by Table 11, become smaller as the mesh size becomes smaller The exact solution at the time t ¼ 0:1 on the interval [À1, 1] and its approximations corresponding to h ¼ 1=250 and h ¼ 1=1000 are displayed in Fig 12 We note that as shown by Fig 12, there are oscillations in the approximation of the velocity around the stationary shocks when the mesh sizes is h ¼ 1=250 However, when the mesh sizes are decreasing, these oscillations are significantly reduced and they eventually vanish for the mesh size h ¼ 1=1000 and for smaller mesh sizes This suggests that the Godunov-type scheme can still work well for this very challenging resonant case 5.7 Test In this test we approximate a Riemann solution of Construction B3 Let the initial data be given by U L ẳ pL ; uL ; aL ị ẳ 8; 1:5; 1:5ị G2 ; U R ẳ pR ; uR ; aR ị ẳ 6; 1; 2:5ị G2 : 5:7ị We have U2 U R ; U 0ỵ Þ ¼ 3:289312074760908 > 0; U2 ðU R ; U ị ẳ 1:558327382179742 < 0: Therefore, according to Construction B3, the exact solution is a 1-rarefaction wave from U L to U , followed by a stationary wave from U to U 2 W 13 ðU L Þ \ W B2 ðU R Þ, then followed by a 2-shock wave from U to U R , where U ; U are reported in Table 12 The errors for Test are reported in the Table 13 The exact solution at the time t ¼ 0:1 on the interval [À1, 1] and its approximations corresponding to h ¼ 1=250 and h ¼ 1=6000 are displayed in Fig 13 One can see from this figure that the approximate solutions are closed and get closer to the exact solution when the mesh size gets smaller Moreover, Table 13 shows that the errors become smaller as the mesh size gets smaller Thus, Test demonstrates the convergence of the approximate solutions by the Godunov-type scheme to the exact solution when the initial data belong to the subsonic region Table 12 States that separate the elementary waves of the exact solution of the Riemann problem in Test p u a U1 U2 6.349907316450684 À1.236055537773879 1.5 8.315568511366481 À0.626594221204599 2.5 Table 13 Errors of numerical approximations for different mesh sizes for Test N kU h À UkL1 kU h À UkL1 =kUkL1 250 500 1000 2000 4000 6000 0.103 0.067 0.039 0.021 0.012 0.0088 0.0064 0.0041 0.0024 0.0013 0.0007 0.00055 628 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Fig 13 Test Exact solution and approximate solutions for different mesh sizes Conclusions and discussions In this paper we build up a Godunov-type scheme that is quasi-conservative The scheme is well-balanced, since it can capture the stationary solutions We present many numerical tests which demonstrate the robustness and the accuracy of the scheme for every form of the exact solutions Tests therefore cover the case where the initial data belong to the supersonic region, or subsonic region, or both All the tests show that the approximate solutions by the scheme converge to the exact solutions The Godunov-type scheme therefore can resolve the difficulty of the well-balanced scheme in [19] when dealing with the resonant cases Moreover, our convergence result is stronger than the one in [22], where a non-convergence case was obtained in the resonant regime A development of this work for the nonisentropic fluids is under our study Acknowledgement This research is funded by Vietnam National University-HCMC D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 629 References [1] A Ambroso, C Chalons, F Coquel, T Galié, Relaxation and numerical approximation of a two-fluid two-pressure 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(1.2) with the initial data of the form 624 D.H Cuong, M.D Thanh / Applied Mathematics and Computation 256 (2015) 602–629 Table States that separate the elementary waves of the exact solution of the. .. Godunov-type scheme for the shallow water equations with variable topography, we aim to build in this paper a Godunov-type scheme for the model of an isentropic fluid in a nozzle with variable cross-section. .. states on both sides of a stationary wave satisfying (2.14) Set l¼ 2jc : cÀ1 Assuming that U and a are fixed It follows from (2.14) that the curve W ðU Þ of stationary waves consisting of all the