DSpace at VNU: Numerical treatment of nonconservative terms in resonant regime for fluid flows in a nozzle with variable cross-section

DSpace at VNU: Numerical treatment of nonconservative terms in resonant regime for fluid flows in a nozzle with variable...

Numerical treatment of nonconservative terms in resonant regime for fluid flows

in a nozzle with variable cross-section

Mai Duc Thanha,⇑, Dietmar Krönerb

a

Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam

b

Institute of Applied Mathematics, University of Freiburg, Hermann-Herder Str 10, 79104 Freiburg, Germany

a r t i c l e i n f o

Article history:

Received 4 October 2011

Received in revised form 2 June 2012

Accepted 19 June 2012

Available online 2 July 2012

Keywords:

Numerical treatment

Well-balanced scheme

Fluid dynamics

Nozzle

Hyperbolic conservation law

Source term

Shock wave

Stationary wave

a b s t r a c t

When data are on both sides of the resonant surface, existing numerical schemes often give unsatisfac-tory results This phenomenon is probably caused by the truncation errors, which are added up to states near the resonant surface that could shift the approximate states into a wrong side of the resonant sur-face In this paper, we enhance the well-balanced scheme constructed in an earlier work with a comput-ing corrector in the computcomput-ing algorithm that selects the admissible equilibrium state We build up two computing correctors of different types: one depends on the mesh-size and the other depends on the time iteration number Each of these correctors will help the algorithm select the correct equilibrium state when there are two possible states Moreover, we also improve the computational method solving the nonlinear equation that determines the equilibrium states by driving an equivalent form of the equa-tion such that the Newton–Raphson method can work perfectly Numerical tests show that our well-bal-anced scheme equipped with each of the above two computing correctors gives good approximations for initial data in resonant regime

Ó 2012 Elsevier Ltd All rights reserved

1 Introduction

We are interested in the numerical treatment of the

nonconser-vative term of the following model of fluid flows in a nozzle with

variable cross-section

@tðaqÞ þ @xðaquÞ ¼ 0;

@tðaquÞ þ @xðaðqu2þ pÞÞ ¼ p@xa; ð1:1Þ

@tðaqeÞ þ @xðauðqe þ pÞÞ ¼ 0; x 2 R; t > 0;

where a ¼ aðxÞ; x 2 R represents the cross-section,qis the density,

u is the velocity,eis the internal energy, T is the pressure, S is the

entropy, and e ¼eþ u2=2 is the total energy A standard way to

put the system(1.1)under the framework of hyperbolic

conserva-tion laws is to supplement it with an addiconserva-tional trivial equaconserva-tion

see[26,27] In the literature, numerical treatments of

nonconserva-tive systems such as(1.1)have attracted lots of attentions of

scien-tists Most existing schemes could succeed to approximate the exact

solutions in strictly hyperbolic domains In particular, in [24] we

build a well-balanced numerical scheme that can capture

equilib-rium states and provides us with good approximations for data in strictly hyperbolic domains In that work, the nonconservative term

is made absorbed by admissible stationary contacts that result equi-librium states See the references therein for related works How-ever, when data are on both sides of the resonant surface at which the system fails to be strictly hyperbolic, numerical oscillations and divergence could be observed For example, when a rarefaction wave

in one side is attached to a stationary contact that jumps to the other side By investigating the selection procedure which chooses the admissible equilibrium point resulted by a stationary wave at the resonance surface, we discover that a computing selection procedure may be different from the theoretical procedure, proba-bly due to the propagation of errors More precisely, errors adding

to a state belonging to one side of the resonant surface may result an approximate state that falls into the other side of the resonant surface

As well-known, the most complicated situation of the system(1.1)

occurs across the resonant surface, where the Riemann problem may admit one, two or three solutions of different structures, see

[38] Consequently, the well-balanced scheme would unable to pro-duce a good approximation the exact solution when errors propa-gate across the resonant surface To deal with the above problem,

we enforce our well-balanced scheme in[24]with a computing correc-tor that enables the algorithm computing the admissible state across the resonant surface to select the right state We will present in this paper two computing correctors of different types: one corrector depends on the mesh-size and the other one depends on the time 0045-7930/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved.

⇑Corresponding author.

E-mail addresses: mdthanh@hcmiu.edu.vn (M.D Thanh), dietmar@mathematik.

uni-freiburg.de (D Kröner).

Contents lists available atSciVerse ScienceDirect

Computers & Fluids

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p fl u i d

iteration number This work is motivated from the analysis of a

sit-uation of a Riemann solution where a rarefaction wave in one side

of the resonant surface reaches the resonant surface and is then

fol-lowed up by a stationary contact that jumps to the other side of the

resonant surface This is the most challenging situation since small

errors may result huge impact for numerical approximation In all

other situations, even when data are on both sides of the resonant

surface, small errors do not play any significant role in our

well-bal-anced scheme[24] This is because the approximate state stays in

the same side of the exact state and the selection procedure for

the admissible wave works Therefore, our well-balanced scheme

in [24]works properly Besides, we also develop in this work a

robust numerical method to compute admissible stationary

con-tacts The nonlinear equation for the density of the admissible

sta-tionary contact will then be transformed into a convex form so that

the Newton–Raphson method works Furthermore, we also describe

an computing algorithm for selecting the admissible stationary

con-tacts Numerical tests show that our well-balanced method after

cooperating one of the above-mentioned two computing correctors

provides us with good approximations of the exact solutions of(1.1)

for data on both side of the resonance surface Moreover, in the

re-cent interesting work[33], by presenting a systematic comparison

of admissible configurations between the one-dimensional

noncon-servative model and the axisymmetric connoncon-servative Euler system,

the authors conclude that there is a very good correspondence

between the two models when the solutions of the axisymmetric

model possesses straight longitudinal shocks, so that no noticeable

transversal shock perturbs the solution Therefore, we also include

several tests where the exact solutions were considered through

the comparison in[33]

There have been many works concerning the model(1.1)in the

literature First, the model(1.1)can theoretically be understood in

the sense of nonconservative products, see [11] The analysis of

shock waves and other waves of(1.1), and related models can be

seen in [27,31,28,38,20,19,15,2,3,29] Numerical approximations

for the model of fluid flows in a nozzle with variable cross-section

were studied in[24,23,33,21,22] Well-balanced schemes for

shal-low water equations was considered by an early work[17], and

then developed in[8,39,21,22,14,34,30] Well-balanced numerical

schemes for a single conservation law with source term were

stud-ied in[18,6,7,16,4] Well-balanced schemes for multi-phase flows

and other models were studied in [5,25,36,1,40–42] Numerical

schemes for nonconservative hyperbolic systems were considered

in[32,37,9,35,12,13,10] See also the references therein

The organization of this paper is as follows In Section2we

pro-vides basic properties of the model(1.1) Section3is devoted to

equilibrium states, where characterization of roots of the nonlinear

equations determining equilibrium states are summarized, and the

computing algorithm for the admissible root is given In Section4

we review our well-balanced scheme and introduce two

comput-ing correctors Section5is devoted to numerical tests Finally in

Section 6 we will draw conclusions on our results and we also

address some future related study

2 Preliminaries

Let us consider a polytropic fluid where the equation of state is

given in the form

p ¼ pðq;SÞ ¼ ðc 1Þ exp S  S

Cv

qc;

wherec>1; Cv>0 and Sare constants Then

pqðq;SÞ ¼cpðq;SÞ

q ; pSðq;SÞ ¼pðq;SÞ

Cv :

Take the variable U ¼ ðq;u; S; aÞ The system(1.1), (1.2)can be written in the vector form

where

AðUÞ ¼

u q 0 u q

a

pq

q u p S

0 0 u 0

0 0 0 0

0 B B

@

1 C C

The matrix AðUÞ in(2.2)admits four real eigenvalues,

k0¼ 0; k1¼ u  c; k2¼ u; k3¼ u þ c; ð2:3Þ

where c is the local sound speed

c ¼ ffiffiffiffiffiffip

q

p

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cpðq;SÞ=q

p

:

The corresponding eigenvectors of AðUÞ can be chosen as

r0¼

u2q

upq

0 aðpq u2Þ

0 B B

@

1 C C

A; r1¼

q

c 0 0

0 B B

1 C

C; r2¼

pS

0

pq 0

0 B B

1 C

C; r3¼

q

c 0 0

0 B B

1 C

C:

Since the characteristic field associated with k0 may coincide with any other field, the system(2.1)is not strictly hyperbolic Set

G1¼ fU : k0ðUÞ < k1ðUÞ < k2ðUÞ < k3ðUÞg;

G2¼ fU : k1ðUÞ < k0ðUÞ < k2ðUÞ < k3ðUÞg;

G3¼ fU : k1ðUÞ < k2ðUÞ < k0ðUÞ < k3ðUÞg;

G4¼ fU : k1ðUÞ < k2ðUÞ < k3ðUÞ < k0ðUÞg:

Rþ¼ fU : k1ðUÞ ¼ k0ðUÞg;

R0¼ fU : k2ðUÞ ¼ k0ðUÞg;

R¼ fU : k3ðUÞ ¼ k0ðUÞg;

R¼Rþ[R[R0:

ð2:4Þ

In what follows we will refer toRas the resonant surface

3 Equilibrium states Let us be given a state U0¼ ðq0;u0;a0Þ with a level of cross-sec-tion a0 and another level cross-section a1 As in [24], a state

U1¼ ðq1;u1;a1Þ with the level cross-section a1which can be con-nected with U0via a stationary wave is determined by the system

½aqu ¼ 0;

u2

2 þ hðq;S0Þ

½S ¼ 0;

where h is the specific enthalpy(3.16), which is given as a function

h ¼ hðq;SÞ by

hðq;SÞ ¼cexp S  S

Cv

Set

AðSÞ ¼ ðc 1Þ exp S  S

Cv

; j¼ AðS0Þ; l¼ 2jc

c 1: ð3:3Þ

In[24], we solve forqfrom the nonlinear equation

lqc þ1 u2þlqc 1

0

q2

þ a0u0q0

a

 2

The function on the left-hand side of(3.4), unfortunately, is not

convex Therefore, the Newton–Raphson method may not work

We look for an equivalent form of(3.4)into a convex form

3.1 Characterization of the roots

To characterize the roots of the nonlinear Eq (3.4), we will

rewrite(3.4)in a form that is convenient for investigating

proper-ties of these roots Employing the techniques in[28], we transform

(3.4)into the following equivalent form

UðU0;a;qÞ :¼ sgnðu0Þ u2lðqc 1qc 1

0 Þ

qa0u0q0

a ¼ 0 ð3:5Þ

As we will see later on, we can easily investigate properties of

the function on the left-hand side of (3.5) The function

q#UðU0;a;qÞ is defined for

0 6q6qðU0Þ :¼ 1

lu

2þqc1 0

c 1

:

A straightforward calculation shows that

@UðU0;a;qÞ

@q ¼

u2lðqc 1qc1

0 Þ jcqc 1

u2lðqc 1qc 1

0 Þ

Assume, for simplicity, that u0>0 The last expression means

that

@UðU0;a;qÞ

@q >0; q<qmaxðU0Þ;

@UðU0;a;qÞ

@q <0; q>qmaxðU0Þ;

where

qmaxðU0Þ :¼ 2

lðcþ 1Þ u

2

þlqc 1 0

c 1

The function q#UðU0;a;qÞ takes negative values at the

endpoints Thus, it admits some root if and only if the maximum

value is non-negative This is equivalent to saying that

a P aminðU0Þ :¼ a0q0ju0j

ffiffiffiffiffiffi

jc

p

qcþ12

maxðU0Þ

For u0<0, similar properties hold Thus, given U0, a stationary

shock issuing from U0and connecting to some state U ¼ ðq;u; aÞ

exists if and only if a P aminðU0Þ When a > aminðU0Þ, then there

are exactly two valuesu1ðU0;aÞ <qmaxðU0Þ <u2ðU0;aÞ such that

UðU0;a;u1ðU0;aÞÞ ¼UðU0;a;u2ðU0;aÞÞ ¼ 0: ð3:8Þ

As in[28], we obtain:

qmaxðU0Þ >q0; ðU0Þ 2 G1[ G4;

qmaxðU0Þ <q0; ðU0Þ 2 G2[ G3; ð3:9Þ

qmaxðU0Þ ¼q0; ðU0Þ 2 C:

The state ðu1ðU0;aÞ; a0u0q0=ðau1ðU0;aÞÞÞ from the other side of

a stationary jump from U0belongs to G1if u0>0, and belongs to G4

if u0<0, while the state ðu2ðU0;aÞ; a0u0q0=ðau2ðU0;aÞÞÞ belongs to

G2if u0>0 and belongs to G3if u0<0 In addition, it holds that

(i) If a > a0, then

u1ðU0;aÞ <q0<u2ðU0;aÞ: ð3:10Þ

(ii) If a < a, then

q0<u1ðU0;aÞ for U02 G1[ G4;

q0>u2ðU0;aÞ for U02 G2[ G3: ð3:11Þ

To select a unique physical root among the two possible roots,

we need the following criterion

ADMISSIBILITY CRITERION.Along the stationary curve between left- and right-hand states of any stationary wave, the component a expressed

as a function ofqhas to be monotone inq

As shown in[38], the above Admissibility Criterion is equivalent

to the condition that any stationary wave has to remain in the closure

of a strictly hyperbolic domain Therefore, for U02 G1[ G4, we chooseu1ðU0;aÞ, and for U02 G2[ G3, we takeu2ðU0;aÞ, where

U0plays the role of a left-hand side state of the stationary contact 3.2 Computing algorithm

In this subsection we will describe the method to compute the admissible root among the two rootsuiðU0;aÞ; i ¼ 1; 2 defined by

(3.8)for given U0and a

The functionq#UðU0;a;qÞ in(3.5)is not convex So we look for an equivalent form of (3.4) such that the resulted equation can be treated numerically by a standard favorite method such

as the Newton–Raphson method Multiplying both sides of(3.4)

by 1=q, we obtain

FðU0;a;qÞ :¼lqc u2þlqc 1

0

qþ a0u0q0

a

 21

q¼ 0; q>0;

ð3:12Þ

wherelis defined by(3.3) Since Eq.(3.12)is an equivalent form of

Eq.(3.5), it has the same roots under the same conditions as seen in the above argument The interesting is that the function on the left-hand side of(3.12)is strictly convex This enable us to apply the Newton–Raphson method to calculate its two rootsu1ðU0;aÞ and

u2ðU0;aÞ Indeed, a simple calculation gives

dFðU0;a;qÞ

dq ¼lcq

c 1 u2þlqc 1

0

 a0u0q0

a

 21

q2;

d2FðU0;a;qÞ

dq2 ¼lcðc 1Þqc 2þ 2 a0u0q0

a

 2 1

q3>0; q>0:

ð3:13Þ

The functionq# FðU0;a;qÞ attains a unique strictly minimum value at a point where its derivative given by(3.13)vanishes How-ever, the analytic form of this minimum point is not available This raises a difficulty when applying the Newton–Raphson method, since we would not know which roots the method gives if we start the method at an arbitrary point In the following we will deal with the starting point of the Newton–Raphson method such that it will give us the admissible root

First, it is not difficult to check that

FðU0;a;qÞUðU0;a;qÞ < 0; 0 <q–uiðU0;aÞ; i ¼ 1; 2: ð3:14Þ

As observed in the previous subsection, we have two roots with

FðU0;a;q0Þ < 0; u1ðU0;aÞ <q0<u2ðU0;aÞ:

To get the root u1ðU0;aÞ, we can use the Newton–Raphson method applied to the function q# FðU0;a;qÞ with a starting point less thanu1ðU0;aÞ How to choose such a point? Consider

’’small’’ values ofq It follows from(3.14)that

FðU0;a;qÞ >  u2þlqc 1

0

qþ a0u0q0

a

 21

qP0 for

q6q1:¼ a0u0q0

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2þlqc 1

0

q <u1ðU0;aÞ: ð3:15Þ

Since

FðU0;a;qÞ > 0; d2FðU0;a;qÞ=dq2>0;

for 0 <q<u1ðU0;aÞ and q1<u1ðU0;aÞ, the Newton–Raphson

method starting at q1will generate a monotone increasing sequence

that converges tou1ðU0;aÞ

In a similar manner, to get the rootu2ðU0;aÞ, we can use the

Newton–Raphson method with a starting point larger than

u2ðU0;aÞ To choose such a starting point, consider ‘‘large’’ values

ofq It follows from(3.14)that

FðU0;a;qÞ >lqc u2þlqc 1

0

qP0

for

qPq2:¼ u

2

lþq

c 1 0

 1=ð c 1Þ

¼ cþ 1 2

 1=ð c 1Þ

qmaxðU0;aÞ >u2ðU0;aÞ:

ð3:16Þ

Since

FðU0;a;qÞ > 0; d2FðU0;a;qÞ=dq2>0;

forq>u2ðU0;aÞ and q2>u2ðU0;aÞ, the Newton–Raphson method

starting at q2will generate a monotone decreasing sequence that

converges tou2ðU0;aÞ

We can summarize the above argument in the following

algorithm which describes the choice for a starting point in the

Newton–Raphson method, applying to calculate the admissible

root of the nonlinear Eq.(3.12)

3.2.1 Algorithm of selecting admissible root

A pseudo-code for Newton–Raphson method selecting the

admissible root of Eq.(3.12)can be described as follows We

con-sider only for u0>0, since the case u060 can be treated similarly

Algorithm 1 q¼ SelectingRootðU0;aÞ

If k1ðU0Þ P 0

- Start at q1:q¼ q1

else

- Starting at q2:q¼ q2

end

FðU0;a;qÞ ¼lqc u2þlqc1

0

qþ a0 u 0q0

a

q; while jFðU0;a;qÞj < 1e  12

dFðU 0 ;a;qÞ

dq ¼lcqc1 u2þlqc1

0

 a0 u 0q0

a

q2;

q¼q FðU0 ;a;qÞ

dFðU 0 ;a;qÞ=dq

FðU0;a;qÞ ¼lqc u2þlqc1

0

qþ a0 u 0q0

a

q; end

4 Numerical schemes and computing correctors

In[24], we built a numerical scheme for the model(1.1) Tests

show that this scheme captures exactly equilibrium states and it

provides us with convergence in strictly hyperbolic regions

More-over, it preserves the positivity of the density and possesses the

numerical minimum entropy principle, see[23] As most existing

schemes, its original version may fail to approximate solutions when

data belong to both sides of the resonant surface In particular, the

scheme may not work when a rarefaction wave is attached by a

sta-tionary wave that jumps into the other side of the resonant surface This is because error propagation probably cause the computing algorithm to select the wrong state in the wrong side To deal with this, in this section we will introduce two different computing correctors that enable the scheme to work properly in that case 4.1 Well-balanced numerical scheme

Given a time stepDt > 0 and a spacial mesh sizeDx Set

xj¼ jDx; j 2 Z; tn¼ nDt; n 2 N; k¼Dt

Dx; ð4:1Þ

Let g ¼ gðU; VÞ be an underlying numerical flux of the usual gas dynamics equations, which corresponds to the case a  constant in

(1.1) Our well-balanced scheme for(1.1)is defined by

Unþ1j ¼ Unj  k gðU nj;Unjþ1;Þ  gðUnj1;þ;UnjÞ

In the scheme(4.2), the states

Un jþ1;¼ ðq;qu;qeÞnjþ1;; Un

j1;þ¼ ðq;qu;qeÞnj1;þ

are defined as follows First, observe that the entropy is constant across each stationary jump, we compute qn

jþ1;;un jþ1; from the equations

an jþ1qn jþ1un jþ1¼ an

jqn jþ1;un jþ1;;ðun jþ1Þ2

2 þ hðqn

jþ1Þ

¼ðu

n jþ1;Þ2

2 þ hðqn

and we computeqn

j1;þ;un j1;þfrom the equations

an j1qn j1un j1¼ an

jqn j1;þun j1;þ;ðun j1Þ2

2 þ hðqn

j1Þ

¼ðu

n j1;þÞ2

2 þ hðqn

It was shown in our earlier work[24]that our scheme((4.1)– (4.4))is well-balanced For example, we can take the Lax–Fried-richs numerical flux:

gðU; VÞ ¼1

2ðf ðUÞ þ f ðVÞÞ 

1 2kðV  UÞ: ð4:5Þ

4.2 Computing correctors

As indicated above, the selection of the states Unj;involve the Monotone Criterion So, when a rarefaction wave started at

UL2 G2approaches and reaches the resonant surfaceRþat a state

U1, it can be followed up by a stationary contact that is attached to

it at U1and jumps into U22 G1 The attaching condition means that the characteristic speed k1ðU1Þ at the end of the rarefaction fan coincides with the discontinuity speed k0ðU1;U2Þ ¼ 0

Let Unj be an approximation of such an above mentioned state

U1of a rarefaction wave from UL2 G2to U12Rþ We need to con-trol the distance between Un

j and the resonant surfaceRþ using

k1ðUnjÞ More precisely, whenever the following condition holds

k1ðUnjÞ < dnj;

where dnj >0 represents a computing corrector, we will take the root

u1ðUnj;ajþ1Þ to jump to Unjþ1;2 G1 Note that in this case Unj may be-long to G1or G2, orRþ In the following we suggest two computing correctors:

(I) A mesh-size dependent corrector:

dnj ¼Dxmax

i¼1;2;3jkiðUnjÞj jqn

jþ1qn

jj þ jun jþ1 un

jj þ jpn jþ1 pn

jj

; ð4:6Þ

(II) Corrector depends on the number of the iterations:

dnj ¼maxi¼1;2;3jkiðU

n

jÞj ffiffiffi k

p jqn

jþ1qn

jj þ junjþ1 unjj þ jpnjþ1 pnjj

; ð4:7Þ

where k is the number of iterations

The way to take into account one of the above computing

cor-rectors can be described in the following algorithm

4.2.1 Algorithm of computing admissible root

A pseudo-code for Newton–Raphson method selecting the

admissible root of the Eq.(3.12)can be described as follows We

consider only for u0>0, since the case u060 can be treated

similarly

Algorithm 2 q¼ ComputingCorrectorðU0;a; dÞ

If k1ðU0Þ P d

– Start at q1:q¼ q1

else

– Starting at q2:q¼ q2

end

FðU0;a;qÞ ¼lqc u2þlqc1

0

qþ a0 u 0q0

a

q; while jFðU0;a;qÞj < 1e  12

dFðU 0 ;a;qÞ

dq ¼lcqc1 u2þlqc1

0

 a0 u 0q0

a

q2;

q¼q FðU0 ;a;qÞ

dFðU 0 ;a;qÞ=dq

FðU0;a;qÞ ¼lqc u2þlqc1

0

qþ a0 u 0q0

a

q; end

5 Test cases

5.1 Test 1

For Tests 1–3 below, the solution is evaluated for x 2 ½1; 1

with the mesh sizes of 1000 points and 3000 points, and at the

time t ¼ 0:2 We take

C:F:L ¼ 0:5:

The following test is devoted to an isentropic ideal gas, where

the pressure is given by

p ¼jqc; c>1;j>0

and in the sequel, for simplicity we takej¼ 1 The governing

equa-tions of the model of the isentropic fluid in a nozzle with variable

cross-section are given by

@tðaqÞ þ @xðaquÞ ¼ 0;

@tðaquÞ þ @xðaðqu2þ pÞÞ ¼ p@xa; x 2 R; t > 0; ð5:1Þ

Let us recall some basic properties of the model(5.1) The

read-er is refread-erred to [28] for more details Taking the variable

U ¼ ðq;u; aÞ, we can re-write the system(5.1),(1.2)in the form

@tU þ AðUÞ @xU ¼ 0;

where

AðUÞ ¼

u q qu=a

h0ðqÞ u 0

0 0 0

0

@

1 A; hðqÞ ¼ jc

c 1qc 1:

The matrix AðUÞ admits the following three eigenvalues

k0:¼ 0; k1:¼ u  ffiffiffiffiffiffiffiffiffiffiffi

p0ðqÞ

p

; k2:¼ u þ ffiffiffiffiffiffiffiffiffiffiffi

p0ðqÞ

p

;

together with the corresponding right-eigenvectors:

r0:¼

qu

p0ðqÞ

au ap0ð q Þ u

0 B

1 C

A r1:¼

q

 ffiffiffiffiffiffiffiffiffiffiffi

p0ðqÞ p 0

0 B

1 C

A r2:¼

q

ffiffiffiffiffiffiffiffiffiffiffi

p0ðqÞ p 0

0 B

1 C A:

Set

C:u ¼ pffiffiffiffiffiffijc

qc12:

We can see that

k1¼ k0 on Cþ

;

k2¼ k0 on C:

We consider the Riemann problem for (5.1), (1.2) with the Riemann data

U0ðxÞ ¼ UL¼ ðqL;uL;aLÞ ¼ ð3; 0:2; 1Þ 2 G2; x < 0;

UR¼ ðqR;uR;aRÞ ¼ ð1:4; 1:5; 1:1Þ 2 G1; x > 0;

ð5:2Þ

where G1 is the domain where k1ðUÞ > 0, and G2 is the domain where k1ðUÞ < 0 and k2ðUÞ > 0 In this test, the Riemann data are ta-ken on the opposite sides of the resonance curve in the ðq;uÞ-plane Set

U1¼ ð1:3783; 1:2616; 1Þ

U2¼ ð0:97819; 1:6161; 1:1Þ;

U3¼ ð1:2304; 1:3387; 1:1Þ:

ð5:3Þ

The exact solution is a rarefaction wave from ULto U12 Cþ, fol-lowed by a stationary wave from U1to U2, followed by a 1-shock from U2to U3, and then arrives at URby a 3-shock

Without a corrector, the well-balanced method with underlying Lax-Friedrichs scheme does not give a good approximation to the exact solution, seeFig 1

5.2 Test 2

In this test, we use the same data as in Test 1, and we equip our well-balanced scheme by the modified version of computing corrector I in(4.6) by neglecting the term of the pressure The underlying Lax-Friedrichs scheme is chosen This tests shows that our method provides us with a good approximations to the exact solution with 1000 and 3000 mesh points for the interval ½1; 1, seeFig 2

Fig 1 Test 1 Without the corrector, the well-balanced method with underlying

5.3 Test 3

This test is devoted to a nonisentropic fluid, where we take

c¼ 1:4; Cv¼ 1; S¼ 1:

First, let us describe a way of computing exact solutions where

data belong to both sides of the resonant surface and that a

rarefac-tion wave in one side is attached by a stararefac-tionary wave that jumps

into the other side

5.4 Computing the typical Riemann solutions

As indicated in[38], we can construct Riemann solutions by

projecting all the wave curves in the ðp; uÞ-plan In particular, a

Rie-mann solution can begins with a 1-rarefaction wave from

UL¼ ðpL;uL;SL;aLÞ 2 G2 and lasts until the rarefaction touches the

resonant surfaceRþat a state U1¼ ðp1;u1;SL;aLÞ since the entropy

is a Riemann invariant At U1, the characteristic speed k1ðU1Þ ¼ 0

and the solution can use a stationary contact to jump to a state

U2¼ ðp2;u2;SL;aRÞ 2 G1 The intersection in the ðp; uÞ-plan of the

forward 1-wave curve W1ðU2Þ and the backward 3-wave curve

W3ðURÞ consists of one state U3 The Riemann solution is thus

con-tinued by a 1-wave from U2to U3 This wave is a 1-shock if p2<p3

and a 1-rarefaction wave otherwise In the computing strategy

be-low we will choose a shock This 1-wave is folbe-lowed by a 2-contact

from U3to U4¼ ðp3;u3;S4;aRÞ

The solution then arrives at URby a 3-wave from U4 This wave

is a 3-shock if p3>pRand a 3-rarefaction wave otherwise In the

sequel we will choose a shock Thus, the solution has the form

Fig 3

R1ðUL;U1Þ ! W0ðU1;U2Þ ! S1ðU2;U3Þ ! W2ðU3;U4Þ ! S3ðU4;URÞ;

ð5:4Þ

where R1ðUL;U1Þ stands for a 1-rarefaction wave from UL to

U1;W0ðU1;U2Þ stands for a stationary contact from U1 to

U2;S1ðU2;U3Þ stands for a 1-shock from U2to U3;W2ðU3;U4Þ stands for a 2-contact from U3to U4, and S3ðU4;URÞ stands for a 3-shock from U4 to UR The computing strategy of the states

Ui;i ¼ 1; 2; 3; 4 can be shown bellow

Setting

mðSÞ ¼c1=2ðc 1Þ1=2cexpS  S

2cCv ; nðSÞ ¼

2mðSÞ

c 1; w ¼

c 1

2c ;

ð5:5Þ

we can rewrite the resonant surfaceRþas

and the 1-wave rarefaction curve R1ðU0Þ as (see[38])

R1ðU0Þ : u ¼ u0 nðS0Þðpw pw

0Þ; p 6 p0: ð5:7Þ

The state U1satisfies the equation

u1¼ mðSLÞpw

1¼ uL nðSLÞðpw

1 pw

which gives

p1¼ uLþ nðSLÞp

w L

mðSLÞ þ nðSLÞ

So, the state U1¼ ðp1;u1;SL;aLÞ is determined by(5.9)and then

(5.8) Next, to evaluate U2, we rewrite the state U1 into the form

U1¼ ðq1;u1;SL;aLÞ with theq-component instead of the p-compo-nent using the equation of state

q¼qðp; SÞ ¼ p

c 1exp

S S

Cv

:

The state U2¼ ðq2;u2;SL;aRÞ is obtained by a stationary contact from U1 As indicated earlier, we calculate theq-component of U2

using the Newton–Raphson method The u-component of U2is fol-lowed immediately

Calculations for the Riemann solution continue with the deter-mination of U3 as follows We rewrite U2 in the form

U2¼ ðp2;u2;SL;aRÞ In the ðp; uÞ-plane, the intersection of S1ðU2Þ and S3ðURÞ determines U3 It is necessary that uR<u2 As shown

in[38], S ðU Þ and S ðU Þ are given by Fig 2 Test 2 The well-balanced method with underlying Lax-Friedrichs scheme equipped by the corrector I in (4.6) gives a good approximation to the exact velocity.

Fig 3 Test 3 The configuration of the exact Riemann solution in the ðx; tÞ-plane.

S1ðU0Þ : u ¼ u0 ðp  p0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞv0

p þlp0

s

; p P p0;

S3ðU0Þ : u ¼ u0þ ðp  p0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞv0

p þlp0

s

; p P p0; l¼c 1

cþ 1: ð5:10Þ

Thus, the state U3satisfies the equations

u ¼ u2 ðp  p2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞv2

p þlp2 s

¼ uRþ ðp  pRÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞvR

p þlpR

s

; p P maxfp2;pRg; ð5:11Þ

wherev¼ 1=q It is derived from(5.11)that the p-component of U3

is the root of the nonlinear equation

f1ðpÞ ¼ ðp  pRÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 lÞvR

p þlpR

s

þ ðp  p2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞv2

p þlp2

s

þ uR u2¼ 0:

ð5:12Þ

It is easy to see that the function f1ðpÞ in (5.12) is strictly increasing and strictly concave So, the root p ¼ p3of the nonlinear

Eq.(5.12)can also be calculated using the Newton–Raphson

meth-od, where

f0

1ðpÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞvR

p

2 ðp þlpRÞ1=2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞvR

p

2 ð1 þlÞpRðp þlpRÞ3=2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞv2

p

2 ðp þlp2Þ1=2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞv2

p

2 ð1 þlÞp2ðp þlp2Þ3=2>0:

Let us now consider the Riemann problem for(1.1),(1.2)with the Riemann data

U0ðxÞ ¼ UL¼ ðqL;uL;pL;aLÞ ¼ ð5; 0:5; 8; 1Þ 2 G2; x < 0;

UR¼ ðqR;uR;pR;aRÞ ¼ ð1; 0:8; 1; 1:2Þ 2 G1; x > 0:

ð5:13Þ

In this test, the Riemann data are taken on the opposite sides of the resonance surface, where UL2 G2and UR2 G1 The solution has the form(5.4), where the states that separate elementary waves of the Riemann solution are given inTable 1

Our well-balanced scheme((4.1)–(4.5))equipped by either Cor-rector I in(4.6)or Corrector II in(4.7)gives good approximations as indicated inFig 4

5.5 Test 4

In this test, we consider a very interesting case where the exact Riemann solution may contain three waves of the same zero speed The states that determine the elementary waves of the exact

Table 1

States that separate the elementary waves of the exact Riemann solution in Test 3, see

Fig 3

Riemann solution are given by Table 17 in[33], where the authors

compare the exact Riemann solution with approximate solutions

obtained from the one-dimensional and the three-dimensional

models For the sake of completeness, we list these states in

Table 2

The exact Riemann solution starts by a stationary wave from UL

to U1, followed by a 1-shock with zero speed from U1to U2, then

followed by another stationary wave from U2to U3 It is attached

by a rarefaction wave from U3to U4, and it contiues with a

2-con-tact discontinuity from U4 to U5, and finally it reaches UR by a

3-shock SeeFig 5, where the states U1;U2, and U3are distributed

along the t-axis in the ðx; tÞ-plane

Our scheme using Corrector (II) in(4.7)computes the

approxi-mate solution at the time t ¼ 0:2s over the interval ½0; 2 with 6000

mesh points As in [33], the initial discontinuity is located at

x ¼ 0:8 The density and velocity of the exact and approximate

solutions are displayed inFig 6 Fig 6shows that the scheme

can provides us with a good approximation to the exact solution

5.6 Test 5

We consider the approximation of an exact Riemann solution that may also contain three waves of the same zero speed The states that determine the elementary waves of the exact Riemann solution are given by Table 16 in[33], where the authors compare the exact Riemann solution with approximate solutions obtained from the one-dimensional and the three-dimensional models Pre-cisely, these states are given inTable 3

The exact Riemann solution starts by a 1-rarefaction wave from

ULto U1, followed by a 2-contact discontinuity from U1to U2, then followed by a rarefaction wave from U2to U3 It is attached by a stationary wave from U3to U4, followed by a 1-shock wave with zero speed from U4to U5, and finally it reaches URby another sta-tionary wave SeeFig 7, where the states U3;U4, and U5are distrib-uted along the t-axis in the ðx; tÞ-plane

Table 2

States that separate the elementary waves of the exact Riemann solution in Test 4, see

Fig 5

Fig 5 Test 4 The configuration of the exact Riemann solution in the ðx; tÞ-plane.

The states U 1 ;U 2 , and U 3 are distributed along the t-axis.

Fig 6 Test 4 Comparison of the the density and velocity of the exact solution and the ones of the approximate solution by the well-balanced scheme equipped with

Table 3 States that separate the elementary waves of the exact Riemann solution in Test 5, see

Fig 7

Fig 7 Test 5 The configuration of the exact Riemann solution in the ðx; tÞ-plane The states U 3 ; U 4 , and U 5 are distributed along the t-axis.

Our scheme using Corrector (II) in(4.7)computes the approxi-mate solution at the time t ¼ 0:12s over the interval ½0; 2 with

6000 mesh points As in[33], the initial discontinuity is located

at x ¼ 0:8 The density and velocity of the exact and approximate solutions are displayed inFig 8, where we display the solutions over the interval ½0; 1:2 for a better view.Fig 8 shows that the scheme can give a good approximation to the exact solution

5.7 Test 6 let us consider another exact Riemann solution given by Table

16 in[33], where the authors compare the exact Riemann solution with approximate solutions obtained from the one-dimensional and the three-dimensional models The states that determine the elementary waves of the exact Riemann solutions are given in

Table 4 The exact Riemann solution starts by a 1-shock wave from ULto

U1, followed by a 2-contact discontinuity from U1 to U2, then followed by a stationary wave from U2to U3 It continues with a 1-shock with zero speed from U3to U4, followed by another sta-tionary wave from U4to U5, and finally it reaches URby a 3-rarefac-tion wave See Fig 9, where the states U3;U4, and U5 are distributed along the t-axis in the ðx; tÞ-plane

Our scheme using Corrector (II) in(4.7)computes the approxi-mate solution at the time t ¼ 0:15s over the interval ½0; 2 with

6000 mesh points As in[33], the initial discontinuity is located

at x ¼ 0:8 The density and velocity of the exact and approximate solutions are displayed inFig 10in the interval ½0; 1:5.Fig 10also

Fig 8 Test 5 Comparison of the the density and velocity of the exact solution and the ones of the approximate solution by the well-balanced scheme equipped with Corrector (II) in (4.7) over ½0; 2 at the time t ¼ 0:12s.

Table 4

States that separate the elementary waves of the exact Riemann solution in Test 6, see

Fig 9

Fig 9 Test 6 The configuration of the exact Riemann solution in the ðx; tÞ-plane.

The states U 3 ;U 4 , and U 5 are distributed along the t-axis.

Fig 10 Test 6 Comparison of the the density and velocity of the exact solution and the ones of the approximate solution by the well-balanced scheme equipped with

indicates that our scheme can provide a reasonable approximation

to the exact solution

6 Conclusions

Most existing schemes for nonconservative systems or

nonstric-tly hyperbolic systems can approximate the exact solutions only in

strictly hyperbolic domains This work gives a way to treat

numer-ically the nonconservative terms of the model of a fluid in a nozzle

with variable cross-section in the resonant regime where data

belong to both sides of the resonant surface We introduce two

types of computing correctors to ‘‘navigate’’ the scheme to take

the right state Tests show that our well-balanced method

equipped by one of these computing correctors gives good

approx-imations Questions on a general approach and higher-order

schemes are open for further study

Acknowledgments

The authors are grateful to the reviewers for their very

con-structive comments and helpful suggestions

This research is funded by Viet Nam National Foundation for

Science and Technology Development (NAFOSTED) under Grant

No 101.02-2011.36

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