A Godunov-type scheme for the isentropic model of a fluid flowin a nozzle with variable cross-section Dao Huy Cuonga,b, Mai Duc Thanhc,⇑ a Nguyen Huu Cau High School, 07 Nguyen Anh Thu, Tr
Trang 1A Godunov-type scheme for the isentropic model of a fluid flow
in a nozzle with variable cross-section
Dao Huy Cuonga,b, Mai Duc Thanhc,⇑
a
Nguyen Huu Cau High School, 07 Nguyen Anh Thu, Trung Chanh Ward, Hoc Mon District, Ho Chi Minh City, Viet Nam
b
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
of the model, pointing out several interesting properties of the wave curves, and ing specific existence domain for each type of solutions Then, we incorporate localRiemann solutions to build a Godunov-type scheme for the model The scheme isconstructed 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 Fur-thermore, tests also show that our Godunov-type scheme can give a good accuracy fornumerical approximations of exact solutions Our Godunov-type scheme can resolve thedifficulty of other existing schemes for similar models of fluid flows with nonconservativeterms
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1 Introduction
In this paper we study to build a Godunov-type scheme for the numerical approximation of weak solutions to the value problem associated with the following isentropic model of a fluid flow in a nozzle with variable cross-section
initial-@tðaqÞ þ @xðaquÞ ¼ 0;
@tðaquÞ þ @xðaðqu2þ pÞÞ ¼ p@xa; x 2 R; t > 0; ð1:1Þ
whereqðx; tÞ; uðx; tÞ; pðx; tÞ denote the density, particle velocity, and pressure of the fluid, respectively, and a ¼ aðxÞ denotesthe 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 ¼ aðxÞ, numerical discretizations will yield piece-wise constant approximate functions.Therefore, the term p@xa is a nonconservative term, and so the system(1.1)is of nonconservative form, see[9] Numericalapproximations for systems of balance laws with nonconservative terms have been a very interesting, but rather challengingtopic for many authors This is because the standard numerical schemes for systems of conservation laws with usual
http://dx.doi.org/10.1016/j.amc.2015.01.024
⇑ Corresponding author.
E-mail addresses: cuongnhc82@gmail.com (D.H Cuong), mdthanh@hcmiu.edu.vn (M.D Thanh).
Contents lists available atScienceDirect
Applied Mathematics and Computation
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a m c
Trang 2discretizations 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 variabletopography, we aim to build in this paper a Godunov-type scheme for the model of an isentropic fluid in a nozzle withvariable cross-section(1.1) The most interesting result of this paper is that our Godunov-type scheme can resolve thedifficulty of the well-balanced scheme in[19]when dealing with the resonant cases Moreover, the scheme can providebetter convergence results than the ones in[22] for shallow water equations with variable topography Observe that aGodunov-type scheme is based on solutions of the local Riemann problem for(1.1), where, as is well-known, we supplementthe system(1.1)the trivial equation
Note that the system(1.1) and (1.2)can be written in the form of a nonconservative system of balance laws
@tU þ AðUÞ@xU ¼ 0;
for U ¼ ðq;u; aÞ, for example, and AðUÞ 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 thewave curves together with characterizations of the existence domain of each kind of Riemann solutions when the initial databelong to different subsonic or supersonic regions Then, we employ these Riemann solutions to build up a Godunov-typescheme for the model Tests show that our Godunov-type scheme is well-balanced as it can capture steady solutions Fur-thermore, 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, forexample,[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 forthe model of a general fluid in a nozzle with discontinuous cross-section, to[21,22,7,25]for the Riemann problem for theshallow 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 Godunovscheme of systems of conservation laws We note that Godunov-type schemes for various hyperbolic systems of balancelaws in nonconservative form were studied in[17,8,22,2,27,26] Numerical schemes for multi-phase flow models were pre-sented 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 Section2we discuss basic concepts and properties of the system(1.1) and(1.2) Section3is devoted to the revisited Riemann problem In Section4we build a Godunov-type scheme for the model(1.1) Section5is devoted to numerical tests Finally, we provide in Section6several conclusions and discussions
The matrix AðUÞ admits the following three eigenvalues
Trang 3The corresponding eigenvectors can be chosen as
r1ðUÞ ¼
q
ffiffiffiffiffiffiffiffiffiffiffi
p0ðqÞp0
0
B
1CA; r2ðUÞ ¼
q
ffiffiffiffiffiffiffiffiffiffiffi
p0ðqÞp0
0B
1CA; r3ðUÞ ¼
qu
p0ðqÞ
aðu p0ðuqÞÞ
0B
1CA:
It is evident that the first and the third characteristic speeds coincide:
k1ðUÞ ¼ k3ðUÞ;
on the upper sonic surface:
Cþ¼ ðq;u; aÞj u ¼ ffiffiffiffiffiffiffiffiffiffiffi
p0ðqÞp
G2¼ fU : k1ðUÞ < 0 < k2ðUÞg ¼ fjuj < ffiffiffiffiffiffiffiffiffiffiffi
p0ðqÞpg;
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 Ul¼ ðql;ul;alÞ; Ur¼ ðqr;ur;arÞ (al¼ ar) is a week tion of the form
Trang 4One simple way to determine the jump relations for shocks of(1.1) and (1.2)was proposed in[20] Briefly, since the shocksatisfy the conservative Eq.(1.2), the usual Rankine–Hugoniot relation
It follows from(2.6)that given a left-hand state U0¼ ðq0;u0;a0Þ, the Hugoniot set HðU0Þ consisting of all right-hand states
U ¼ ðq;u; a0Þ that can be connected to U0by a shock is given by
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 Ulto a right-hand state Urin the genuinely nonlinear characteristicfields satisfy the Lax shock inequalities
kiðUrÞ < kiðUl;UrÞ < kiðUlÞ; i ¼ 1; 2; ð2:7Þ
where kiis the shock speed, i ¼ 1; 2 A Lax shock is a shock which satisfies the Lax shock inequalities(2.7)
We now define the forward curves SiðU0Þ of admissible i-shock waves which consist of all right-hand states U that can beconnected to a given left-hand state U0by an i-Lax shock wave, i ¼ 1; 2 It is not difficult to check that these curves are givenby
The conditionqPq0for S1ðU0Þ (respectivelyq6q0for S2ðU0Þ) is derived from the Lax shock inequalities(2.7)
Similarly, the backward i-shock wave curves SB
iðU0Þ consisting of all left-hand states U that can be connected to a givenright-hand state U0by an i-Lax shock wave, i ¼ 1; 2, are given by
Trang 52.3 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-similarsolution of the form
is a weak solution of(1.1) and (1.2), called an i-rarefaction wave connecting the left-hand state Ulto the right-hand state
Ur;i ¼ 1; 2 We note that Ulis located on the left and Uris located on the right of the rarefaction fan, seeFig 2
Note that as a consequence of the formulas of the eigenvectors r1and r2, we have
Trang 6The backward i-rarefaction wave curve RB
iðU0Þ consisting of all left-hand states U that can be connected to a given right-handstate U0by an i-rarefaction wave, i ¼ 1; 2, are given by
From the above analysis, we can not define the wave curves
W1ðU0Þ ¼ S1ðU0Þ [ R1ðU0Þ;
WBðU0Þ ¼ SBðU0Þ [ RBðU0Þ;
W2ðU0Þ ¼ S2ðU0Þ [ R2ðU0Þ;
WBðU0Þ ¼ SBðU0Þ [ RBðU0Þ:
ð2:11Þ
The above argument shows that the curves WiðU0Þ; WB
iðU0Þ can be parameterized as
Accordingly, for each U ¼ ðq;uÞ, we defineU2ðUR;UÞ as follows:
where the function wBðUR;qÞ is defined as(2.12) Obviously,U2ðUR;UÞ ¼ 0 for U 2 WB
ðURÞ Moreover,U2ðUR;UÞ > 0 for U isabove WBðURÞ andU2ðUR;UÞ < 0 for U is below WBðURÞ
Besides, it is not difficult to check that the wave curve W1ðU0Þ :q# u¼ uðqÞ;q>0 is strictly decreasing, and the wavecurve WBðU0Þ :q# uðqÞ;q>0 is strictly increasing
Assuming that U0and a are fixed It follows from(2.14)that the curve W3ðU0Þ of stationary waves consisting of all the states
U that can be connected to U0by a stationary wave can be parameterized byq Precisely, the curve W3ðU0Þ is given by
W3ðU0Þ : u ¼ w3ðU0;qÞ :¼ sgnðu0Þ u2lðqc 1qc1
Trang 7Eliminating u ¼ w3ðU0;qÞ in(2.15), we obtain the following nonlinear equation inq
Fðu1ðU0;aÞÞ ¼ Fðu2ðU0;aÞÞ ¼ 0: ð2:19Þ
The following lemma characterizes the stationary waves
Lemma 2.1 [20, Lem 2.3] The following conclusions hold
(b) The state ðu1ðU0;aÞ; w3ðU0;u1ðU0;aÞÞ from the other side of a stationary jump from U0belongs to G1if u0>0, and belongs
to G3if u0<0, while the state ðu2ðU0;aÞ; w3ðU0;u2ðU0;aÞÞ belongs to G2 In addition, it holds that
(i) If a > a0, then
u1ðU0;aÞ <q0<u2ðU0;aÞ: ð2:21Þ
(ii) If a < a0, then
q0<u1ðU0;aÞ for U02 G1[ G3;
q0>u2ðU0;aÞ for U02 G2: ð2:22Þ
(c)
a > aminðUÞ; ðq;u; aÞ 2 Gi;i ¼ 1; 2; 3;
a ¼ aminðU; aÞ; ðq;u; aÞ 2 C: ð2:23Þ
It follows fromLemma 2.1that there are two possible stationary waves from a given state U0to a state with a new levelcross-section a Thus, it is necessary to impose some condition to select a unique physical stationary wave as follows
Trang 8(MC) Along the stationary wave curve W3ðU0Þ defined by(2.15), the component a has to be monotone with respect toq.Furthermore, the total variation of the cross-section component of any Riemann solution must not exceed jaR aLj,where aR;aLare 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;plane
uÞ-It is interesting to see that the shock speeds k1; k2may vanish, and therefore can coincide with the zero-shock speed of thestationary shocks, as indicated in the following lemma
k1ðU0;UÞ < 0; U 2 S1ðU0Þ: ð2:24Þ
If U02 G1, then there is exactly one state, denoted by U#
0 2 S1ðU0Þ \ Gþ2, such that
(ii) The 2-shock speed k2ðU0;UÞ may change sign along the backward 2-shock curve SBðU0Þ More precisely, if U02 G1[ G2then
2ðU0;UÞ remains positive:
k2ðU0;UÞ > 0; U 2 SB
If U02 G3, then there is exactly one state U#
02 S2ðU0Þ \ G2, such that
3 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, orRiemann solutions, are made of a finite number of elementary waves, which are Lax shocks, rarefaction waves, or admissiblestationary waves It is sufficient to consider only the Riemann data in G1[ Cþ[ G2;aR>aL, since the other cases can similarly
be obtained The constructions will be based on the left-hand state UL, and we distinguish between two cases:
denotes the state resulting from a zero-speed shock wave from U
(iv) U0denotes the state resulting from a stationary wave from U
3.1 Case A: UL2 G1[ Cþ
In this subsection we construct three composite wave curves W31ðULÞ; W313ðULÞ and W13ðULÞ corresponding to UL2 G1[ Cþ
in order to build three constructions A1, A2 and A3 Then, it is interesting that we can point out a ‘‘large enough’’ hood containing UL such that the Riemann problem for (1.1) and (1.2)admits a solution whenever UR belongs thisneighborhood
Trang 9neighbor-Construction A1 We construct the composite wave curve W31ðULÞ as follows First, the solution begins with a stationarywave from ULto U1¼ U0L using smaller rootu1ðUL;aRÞ to shift aLto aR, where
Second, the next part of the solution is a 1-wave from U1¼ U0Lto a state U 2 W1ðU0LÞ such that 0 6q6q0#
L Last, the posite wave curve W31ðULÞ is defined as
com-W31ðULÞ ¼ fUðq;u; aRÞ : U 2 W1ðU0LÞ; 0 6q6q0#
consequently, the Riemann problem for(1.1) and (1.2)has a solution of the form
W3ðUL;U1Þ W1ðU1;U2Þ W2ðU2;URÞ: ð3:7Þ
Remark 3.1 The solution(3.7) always makes sense Indeed, first, if U2 is below UR on WB
ðURÞ then W2ðU2;URÞ is a rarefaction wave R2ðU2;URÞ Since U22 W31ðULÞ G1[ Cþ[ Gþ2, then URbelongs to G1[ Cþ[ G2; therefore k2ðURÞ > 0 and(3.7)makes sense Second, if U2is above URon WBðURÞ then W2ðU2;URÞ is a 2-shock wave S2ðU2;URÞ; therefore(3.7)alsomakes sense, since U22 G1[ Cþ[ Gþ2 and k2ðUR;U2Þ > 0
2-Construction A2 We construct the composite wave curve W313ðULÞ as follows First, for each cross-section level
aM2 ½aL;aR, the solution begins with a stationary wave from ULto U1¼ UMusing smaller rootu1ðUL;aMÞ to shift aLto aM,where
UM¼ ðqM;uM;aMÞ 2 W3ðULÞ;
Trang 10Second, the next part of the solution is a zero-speed 1-shock from U1¼ UMto state U2¼ U#M, i.e.
U2¼ U#M2 S1ðUMÞ;
k1ðUM;U#
Third, the next part of the solution is a stationary wave from U2¼ U#
Mto state U3¼ U#0M using bigger rootu2ðU2;aRÞ to shift
consequently, the Riemann problem for(1.1) and (1.2)has a solution of the form
W3ðUL;U1Þ S1ðU1;U2Þ W3ðU2;U3Þ W2ðU3;URÞ: ð3:15Þ
Remark 3.2 The solution(3.15)always makes sense since U32 W313ðULÞ Gþ2
Construction A3 We construct the composite wave curve W13ðULÞ as follows First, the solution begins with 1-shock wavefrom ULto a state U1, where
U12 S1ðULÞ;
So, U1is located between U#
L and Uon S1ðULÞ, where
hence, the Riemann problem for(1.1) and (1.2)has a solution of the form
S1ðUL;U1Þ W3ðU1;U2Þ W2ðU2;URÞ; ð3:23Þ
provide that kðU ;U Þ > 0 when W ðU ;U Þ is a 2-shock wave SðU ;U Þ
Trang 11Remark 3.3 Since U2 may belong to G
2, then the solution (3.23) may not make sense By providing a condition
UR2 G1[ Cþ[ G2, we will have k2ðUR;U2Þ > 0 when W2ðU2;URÞ is a 2-shock wave S2ðU2;URÞ and k2ðURÞ > 0 when
W2ðU2;URÞ is a 2-rarefaction wave R2ðU2;URÞ; therefore the solution(3.23)makes sense
It follows from three curves W31ðULÞ; W313ðULÞ and W13ðULÞ that we define a continuous curveCAas:
CA¼ W31ðULÞ [ W313ðULÞ [ W13ðULÞ: ð3:24Þ
Obviously, Uupand U0are two end-points ofCA Based on those, we consider the open set OA:
OA¼ Un ¼ ðq ;u ;aRÞ : U2ðU ;UupÞ:U2ðU ;U0Þ < 0o
It is not difficult to check that OA\ ðG1[ Cþ[ G2Þ is an open set containing UL; hence, this is a ‘‘large enough’’ neighborhood
of UL Obviously, the Riemann problem for(1.1) and (1.2)admits a solution whenever URbelongs to OA\ ðG1[ Cþ[ G2Þ.Accordingly, we can show that the rectangle RAdefined as follows is a subset of OA(seeFig 4)
Indeed, let U belong to RA Since 0 <q <q0
and wBðU ;qÞ is a increasing function inq, we have
Theorem 3.1 Assume that UL2 G1[ Cþand aR>aL Consider the open set OAdefined as(3.25)and the rectangle RAdefined as(3.26) It is hold that:
RAis a subset of OA;
ULbelongs to RA;
if URbelongs to RA\ ðG1[ Cþ[ G2Þ, the Riemann problem for(1.1) and (1.2)has a solution of the form(3.7), or of the form(3.15), or of the form(3.23)
Trang 123.2 Case B: UL2 G2
Similar to case A, in this subsection we construct three composite wave curves W131ðULÞ; W1313ðULÞ and W13ðULÞ sponding to UL2 G2in order to build three constructions B1, B2 and B3 Then, it is interesting that we can point out a ‘‘largeenough’’ open set containing ULsuch that the Riemann problem for(1.1) and (1.2)admits a solution whenever URbelongsthis set
corre-Construction B1 We construct the composite wave curve W131ðULÞ as follows First, the solution begins with a tion wave from ULto U1¼ Uþ, where
Third, the next part of the solution is a 1-wave from U2¼ U1þto a state U 2 W1ðU1þÞ such that 0 6q6q1#
þ Last, the ite wave curve W131ðULÞ is defined as
compos-W131ðULÞ ¼ fUðq;u; aRÞ : U 2 W1ðU1þÞ; 0 6q6q1#
It is easy to check that
and W131ðULÞ is a part of the curve W1ðU1þÞ Obviously, Uþupand U1#
þ are two end-points of W131ðULÞ, where
Uþup¼ ð0; uþup;aRÞ ¼ W131ðULÞ \ fq¼ 0g: ð3:32Þ
SeeFig 5 Therefore, if the following holds
U2ðUR;UþupÞ:U2ðUR;U1#
we will have an intersection
consequently, the Riemann problem for(1.1) and (1.2)has a solution of the form
R1ðUL;U1Þ W3ðU1;U2Þ W1ðU2;U3Þ W2ðU3;URÞ: ð3:35Þ
Trang 13Remark 3.4 Since U32 W131ðULÞ G1[ Cþ[ Gþ2, the solution(3.35)always makes sense.
Construction B2 We construct the composite wave curve W1313ðULÞ as follows First, for each cross-section level
aM2 ½aL;aR, the Riemann solution begins with a 1-rarefaction wave from ULto U1¼ Uþ, where Uþis defined as(3.24) ond, the next part of the solution is a stationary wave from U1¼ Uþto U2¼ UþMusing smaller rootu1ðUþ;aMÞ to shift aLto
hence, the Riemann problem for(1.1) and (1.2)has a solution of the form
R1ðUL;U1Þ W3ðU1;U2Þ S1ðU2;U3Þ W3ðU3;U4Þ W2ðU4;URÞ: ð3:44Þ
Remark 3.5 Since U42 W1313ðULÞ Gþ2, the solution(3.44)always makes sense
Construction B3 We construct the composite wave curve W13ðULÞ as follows First, the solution begins with the 1-wavefrom U to a state U , where
Fig 6 The rectangle R B
Trang 14hence, the Riemann problem for(1.1) and (1.2)has a solution of the form
W1ðUL;U1Þ W3ðU1;U2Þ W2ðU2;URÞ; ð3:51Þ
provide that k2ðUR;U2Þ > 0 when W2ðU2;URÞ is a 2-shock wave S2ðU2;URÞ
Remark 3.6 Since U2 may belong to G
2, then the solution (3.51) may not make sense By providing a condition
UR2 G1[ Cþ[ G2, we will have k2ðUR;U2Þ > 0 when W2ðU2;URÞ is a 2-shock wave S2ðU2;URÞ and k2ðURÞ > 0 when
W2ðU2;URÞ is a 2-rarefaction wave R2ðU2;URÞ; therefore the solution(3.51)makes sense
Similar to theTheorem 3.1, we also have the following theorem
Theorem 3.2 Assume that UL2 G2and aR>aL Consider the open set OBand the rectangle RB(seeFig 6) defined as:
OB¼ U ¼ ðq ;u ;aRÞ : U2ðU ;UþupÞ:U2ðU ;U0
4 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 nov-type scheme As seen later, this numerical scheme has a quasi-conservative property Let us set
0B
1CA; SðUÞ ¼ 1a
qu
qu2
0
0B
1C