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

Computers & Fluids 66 (2012) 130–139

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

Numerical treatment of nonconservative terms in resonant regime for fluid flows
in a nozzle with variable cross-section
Mai Duc Thanh a,⇑, Dietmar Kröner b
a
b

Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam
Institute of Applied Mathematics, University of Freiburg, Hermann-Herder Str. 10, 79104 Freiburg, Germany

a r t i c l e

i n f o

a b s t r a c t

Article history:
Received 4 October 2011
Received in revised form 2 June 2012
Accepted 19 June 2012
Available online 2 July 2012

When data are on both sides of the resonant surface, existing numerical schemes often give unsatisfactory results. This phenomenon is probably caused by the truncation errors, which are added up to states
near the resonant surface that could shift the approximate states into a wrong side of the resonant surface. In this paper, we enhance the well-balanced scheme constructed in an earlier work with a computing corrector in the computing algorithm that selects the admissible equilibrium state. We build up two
computing correctors of different types: one depends on the mesh-size and the other depends on the

time iteration number. Each of these correctors will help the algorithm select the correct equilibrium
state when there are two possible states. Moreover, we also improve the computational method solving
the nonlinear equation that determines the equilibrium states by driving an equivalent form of the equation such that the Newton–Raphson method can work perfectly. Numerical tests show that our well-balanced scheme equipped with each of the above two computing correctors gives good approximations for
initial data in resonant regime.
Ó 2012 Elsevier Ltd. All rights reserved.

Keywords:
Numerical treatment
Well-balanced scheme
Fluid dynamics
Nozzle
Hyperbolic conservation law
Source term
Shock wave
Stationary wave

1. Introduction
We are interested in the numerical treatment of the nonconservative term of the following model of fluid flows in a nozzle with
variable cross-section

@ t ðaqÞ þ @ x ðaquÞ ¼ 0;
@ t ðaquÞ þ @ x ðaðqu2 þ pÞÞ ¼ p@ x a;
@ t ðaqeÞ þ @ x ðauðqe þ pÞÞ ¼ 0;

x 2 R;

ð1:1Þ
t > 0;

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

u is the velocity, e is the internal energy, T is the pressure, S is the
entropy, and e ¼ e þ u2 =2 is the total energy. A standard way to
put the system (1.1) under the framework of hyperbolic conservation laws is to supplement it with an additional trivial equation

@ t a ¼ 0;

ð1:2Þ

see [26,27]. In the literature, numerical treatments of nonconservative systems such as (1.1) have attracted lots of attentions of scientists. Most existing schemes could succeed to approximate the exact
solutions in strictly hyperbolic domains. In particular, in [24] we
build a well-balanced numerical scheme that can capture equilib⇑ Corresponding author.
E-mail addresses: (M.D. Thanh), dietmar@mathematik.
uni-freiburg.de (D. Kröner).
0045-7930/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
/>
rium states and provides us with good approximations for data in
strictly hyperbolic domains. In that work, the nonconservative term
is made absorbed by admissible stationary contacts that result equilibrium states. See the references therein for related works. However, when data are on both sides of the resonant surface at which
the system fails to be strictly hyperbolic, numerical oscillations and
divergence could be observed. For example, when a rarefaction wave
in one side is attached to a stationary contact that jumps to the
other side. By investigating the selection procedure which chooses
the admissible equilibrium point resulted by a stationary wave at
the resonance surface, we discover that a computing selection
procedure may be different from the theoretical procedure, probably due to the propagation of errors. More precisely, errors adding
to a state belonging to one side of the resonant surface may result an
approximate state that falls into the other side of the resonant surface.
As well-known, the most complicated situation of the system (1.1)
occurs across the resonant surface, where the Riemann problem
may admit one, two or three solutions of different structures, see

[38]. Consequently, the well-balanced scheme would unable to produce a good approximation the exact solution when errors propagate across the resonant surface. To deal with the above problem,
we enforce our well-balanced scheme in [24] with a computing corrector that enables the algorithm computing the admissible state across
the resonant surface to select the right state. We will present in this
paper two computing correctors of different types: one corrector
depends on the mesh-size and the other one depends on the time


131

M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139

iteration number. This work is motivated from the analysis of a situation of a Riemann solution where a rarefaction wave in one side
of the resonant surface reaches the resonant surface and is then followed up by a stationary contact that jumps to the other side of the
resonant surface. This is the most challenging situation since small
errors may result huge impact for numerical approximation. In all
other situations, even when data are on both sides of the resonant
surface, small errors do not play any significant role in our well-balanced scheme [24]. This is because the approximate state stays in
the same side of the exact state and the selection procedure for
the admissible wave works. Therefore, our well-balanced scheme
in [24] works properly. Besides, we also develop in this work a
robust numerical method to compute admissible stationary contacts. The nonlinear equation for the density of the admissible stationary contact will then be transformed into a convex form so that
the Newton–Raphson method works. Furthermore, we also describe
an computing algorithm for selecting the admissible stationary contacts. Numerical tests show that our well-balanced method after
cooperating one of the above-mentioned two computing correctors
provides us with good approximations of the exact solutions of (1.1)
for data on both side of the resonance surface. Moreover, in the recent interesting work [33], by presenting a systematic comparison
of admissible configurations between the one-dimensional nonconservative model and the axisymmetric conservative Euler system,
the authors conclude that there is a very good correspondence
between the two models when the solutions of the axisymmetric
model possesses straight longitudinal shocks, so that no noticeable

transversal shock perturbs the solution. Therefore, we also include
several tests where the exact solutions were considered through
the comparison in [33].
There have been many works concerning the model (1.1) in the
literature. First, the model (1.1) can theoretically be understood in
the sense of nonconservative products, see [11]. The analysis of
shock waves and other waves of (1.1), and related models can be
seen in [27,31,28,38,20,19,15,2,3,29]. Numerical approximations
for the model of fluid flows in a nozzle with variable cross-section
were studied in [24,23,33,21,22]. Well-balanced schemes for shallow water equations was considered by an early work [17], and
then developed in [8,39,21,22,14,34,30]. Well-balanced numerical
schemes for a single conservation law with source term were studied in [18,6,7,16,4]. Well-balanced schemes for multi-phase flows
and other models were studied in [5,25,36,1,40–42]. Numerical
schemes for nonconservative hyperbolic systems were considered
in [32,37,9,35,12,13,10]. See also the references therein.
The organization of this paper is as follows. In Section 2 we provides basic properties of the model (1.1). Section 3 is devoted to
equilibrium states, where characterization of roots of the nonlinear
equations determining equilibrium states are summarized, and the
computing algorithm for the admissible root is given. In Section 4
we review our well-balanced scheme and introduce two computing correctors. Section 5 is devoted to numerical tests. Finally in
Section 6 we will draw conclusions on our results and we also
address some future related study.

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

U t þ AðUÞU x ¼ 0;
where

0

u

B pq
Bq
AðUÞ ¼ B
B
@0
0

q 0
u

pS

0

u

0

0

k0 ¼ 0;

where c > 1; C v > 0 and SÃ are constants. Then

pq ðq; SÞ ¼

cpðq; SÞ

pðq; SÞ
; pS ðq; SÞ ¼
:
Cv
q

1

C
0C
C:
C
0A
0

k1 ¼ u À c;

ð2:2Þ

k2 ¼ u;

k3 ¼ u þ c;

ð2:3Þ

where c is the local sound speed

c¼

pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pq ¼ cpðq; SÞ=q:

The corresponding eigenvectors of AðUÞ can be chosen as

0
B
B
r0 ¼ B
B
@

u2 q
Àupq
0
aðpq À u2 Þ

1

0

C
C
C;
C
A

B Àc C
C
B
r1 ¼ B

C;
@ 0 A

q

1

1
ÀpS
B 0 C
C
B
r2 ¼ B
C;
@ pq A

0

0 1

0

q

BcC
B C
r 3 ¼ B C:
@0A

0

0

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

G1 ¼ fU : k0 ðUÞ < k1 ðUÞ < k2 ðUÞ < k3 ðUÞg;
G2 ¼ fU : k1 ðUÞ < k0 ðUÞ < k2 ðUÞ < k3 ðUÞg;
G3 ¼ fU : k1 ðUÞ < k2 ðUÞ < k0 ðUÞ < k3 ðUÞg;
G4 ¼ fU : k1 ðUÞ < k2 ðUÞ < k3 ðUÞ < k0 ðUÞg:

ð2:4Þ

Rþ ¼ fU : k1 ðUÞ ¼ k0 ðUÞg;
R0 ¼ fU : k2 ðUÞ ¼ k0 ðUÞg;
RÀ ¼ fU : k3 ðUÞ ¼ k0 ðUÞg;
R ¼ Rþ [ RÀ [ R0 :
In what follows we will refer to R as the resonant surface.
3. Equilibrium states

Let us be given a state U 0 ¼ ðq0 ; u0 ; a0 Þ with a level of cross-section a0 and another level cross-section a1 . As in [24], a state
U 1 ¼ ðq1 ; u1 ; a1 Þ with the level cross-section a1 which can be connected with U 0 via a stationary wave is determined by the system

½aquŠ ¼ 0;

!
u2
þ hðq; S0 Þ ¼ 0;
2

ð3:1Þ

½SŠ ¼ 0;
where h is the specific enthalpy (3.16), which is given as a function
h ¼ hðq; SÞ by

hðq; SÞ ¼ c exp



S À SÃ c
p ¼ pðq; SÞ ¼ ðc À 1Þ exp
q;
Cv

q

uq
a

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

2. Preliminaries
Let us consider a polytropic fluid where the equation of state is
given in the form

ð2:1Þ




S À SÃ cÀ1
q :
Cv

ð3:2Þ

Set

AðSÞ ¼ ðc À 1Þ exp



S À SÃ
;
Cv

j ¼ AðS0 Þ; l ¼

2jc
:
cÀ1

ð3:3Þ

In [24], we solve for q from the nonlinear equation





lqcþ1 À u20 þ lq0cÀ1 q2 þ

a u q 2
0 0 0
¼ 0:
a

ð3:4Þ


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M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139

q0 < u1 ðU 0 ; aÞ for U 0 2 G1 [ G4 ;
q0 > u2 ðU 0 ; aÞ for U 0 2 G2 [ G3 :

The function on the left-hand side of (3.4), unfortunately, is not
convex. Therefore, the Newton–Raphson method may not work.
We look for an equivalent form of (3.4) into a convex form.
3.1. Characterization of the roots
To characterize the roots of the nonlinear Eq. (3.4), we will
rewrite (3.4) in a form that is convenient for investigating properties of these roots. Employing the techniques in [28], we transform
(3.4) into the following equivalent form



UðU 0 ; a; qÞ :¼ sgnðu0 Þ u20 À lðqcÀ1 À qc0À1 Þ

1=2

qÀ

a0 u0 q0
¼0
a
ð3:5Þ

As we will see later on, we can easily investigate properties of
the function on the left-hand side of (3.5). The function
q # UðU 0 ; a; qÞ is defined for

ð3:11Þ

To select a unique physical root among the two possible roots,
we need the following criterion.
ADMISSIBILITY CRITERION. Along the stationary curve between left- and
right-hand states of any stationary wave, the component a expressed
as a function of q has to be monotone in q.
As shown in [38], the above Admissibility Criterion is equivalent
to the condition that any stationary wave has to remain in the closure
of a strictly hyperbolic domain. Therefore, for U 0 2 G1 [ G4 , we
choose u1 ðU 0 ; aÞ, and for U 0 2 G2 [ G3 , we take u2 ðU 0 ; aÞ, where
U 0 plays the role of a left-hand side state of the stationary contact.
3.2. Computing algorithm

@ UðU 0 ; a; qÞ u20 À lðqcÀ1 À q0 Þ À jcqcÀ1
¼ 
1=2 :
@q

u20 À lðqcÀ1 À qc0À1 Þ

In this subsection we will describe the method to compute the
admissible root among the two roots ui ðU 0 ; aÞ; i ¼ 1; 2 defined by
(3.8) for given U 0 and a.
The function q # UðU 0 ; a; qÞ in (3.5) is not convex. So we look
for an equivalent form of (3.4) such that the resulted equation
can be treated numerically by a standard favorite method such
as the Newton–Raphson method. Multiplying both sides of (3.4)
by 1=q, we obtain

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

a u q 2 1


0 0 0
FðU 0 ; a; qÞ :¼ lqc À u20 þ lqc0À1 q þ
¼ 0;
a
q

 ðU 0 Þ :¼
06q6q



1

l

u20

cÀ1

þ q0

1
cÀ1

:

A straightforward calculation shows that
cÀ1

@ UðU 0 ; a; qÞ
> 0;
@q
@ UðU 0 ; a; qÞ
< 0;
@q

ð3:12Þ

q < qmax ðU 0 Þ;
q > qmax ðU 0 Þ;

where

qmax ðU 0 Þ :¼

1

cÀ1
2
cÀ1
:
u20 þ lq0
lðc þ 1Þ



ð3:6Þ

The function q # UðU 0 ; a; qÞ takes negative values at the
endpoints. Thus, it admits some root if and only if the maximum
value is non-negative. This is equivalent to saying that

a P amin ðU 0 Þ :¼

a0 q0 ju0 j
:
pffiffiffiffiffiffi cþ1
2
jcqmax
ðU 0 Þ

ð3:7Þ

For u0 < 0, similar properties hold. Thus, given U 0 , a stationary
shock issuing from U 0 and connecting to some state U ¼ ðq; u; aÞ
exists if and only if a P amin ðU 0 Þ. When a > amin ðU 0 Þ, then there
are exactly two values u1 ðU 0 ; aÞ < qmax ðU 0 Þ < u2 ðU 0 ; aÞ such that

UðU 0 ; a; u1 ðU 0 ; aÞÞ ¼ UðU 0 ; a; u2 ðU 0 ; aÞÞ ¼ 0:

ð3:8Þ

As in [28], we obtain:

qmax ðU 0 Þ > q0 ; ðU 0 Þ 2 G1 [ G4 ;
qmax ðU 0 Þ < q0 ; ðU 0 Þ 2 G2 [ G3 ;
qmax ðU 0 Þ ¼ q0 ; ðU 0 Þ 2 CÆ :


 a u q 2 1
dFðU 0 ; a; qÞ
0 0 0
¼ lcqcÀ1 À u20 þ lqc0À1 À
;
dq
a
q2
2
a u q 2 1
d FðU 0 ; a; qÞ
0 0 0
¼ lcðc À 1ÞqcÀ2 þ 2
> 0; q > 0:

2
dq
a
q3
ð3:13Þ
The function q # FðU 0 ; a; qÞ attains a unique strictly minimum
value at a point where its derivative given by (3.13) vanishes. However, the analytic form of this minimum point is not available. This
raises a difficulty when applying the Newton–Raphson method,
since we would not know which roots the method gives if we start
the method at an arbitrary point. In the following we will deal with
the starting point of the Newton–Raphson method such that it will
give us the admissible root.
First, it is not difficult to check that

FðU 0 ; a; qÞUðU 0 ; a; qÞ < 0;

(i) If a > a0 , then

(ii) If a < a0 , then

where l is defined by (3.3). Since Eq. (3.12) is an equivalent form of
Eq. (3.5), it has the same roots under the same conditions as seen in
the above argument. The interesting is that the function on the lefthand side of (3.12) is strictly convex. This enable us to apply the
Newton–Raphson method to calculate its two roots u1 ðU 0 ; aÞ and
u2 ðU 0 ; aÞ. Indeed, a simple calculation gives

ð3:9Þ

The state ðu1 ðU 0 ; aÞ; a0 u0 q0 =ðau1 ðU 0 ; aÞÞÞ from the other side of
a stationary jump from U 0 belongs to G1 if u0 > 0, and belongs to G4

if u0 < 0, while the state ðu2 ðU 0 ; aÞ; a0 u0 q0 =ðau2 ðU 0 ; aÞÞÞ belongs to
G2 if u0 > 0 and belongs to G3 if u0 < 0. In addition, it holds that

u1 ðU 0 ; aÞ < q0 < u2 ðU 0 ; aÞ:

q > 0;

ð3:10Þ

0 < q – ui ðU 0 ; aÞ;

i ¼ 1; 2:

ð3:14Þ

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

FðU 0 ; a; q0 Þ < 0;

u1 ðU 0 ; aÞ < q0 < u2 ðU 0 ; aÞ:

To get the root u1 ðU 0 ; aÞ, we can use the Newton–Raphson
method applied to the function q # FðU 0 ; a; qÞ with a starting
point less than u1 ðU 0 ; aÞ. How to choose such a point? Consider
’’small’’ values of q. It follows from (3.14) that

a u q 2 1


0 0 0

FðU 0 ; a; qÞ > À u20 þ lqc0À1 q þ
P0
a
q
for


133

M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139

a0 u0 q0
ffi < u1 ðU 0 ; aÞ:
q 6 q1 :¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a u20 þ lq0cÀ1

ð3:15Þ

Since

FðU 0 ; a; qÞ > 0;

2

d FðU 0 ; a; qÞ=dq2 > 0;

for 0 < q < u1 ðU 0 ; aÞ and q1 < u1 ðU 0 ; aÞ, the Newton–Raphson
method starting at q1 will generate a monotone increasing sequence
that converges to u1 ðU 0 ; aÞ.
In a similar manner, to get the root u2 ðU 0 ; aÞ, we can use the

Newton–Raphson method with a starting point larger than
u2 ðU 0 ; aÞ. To choose such a starting point, consider ‘‘large’’ values
of q. It follows from (3.14) that



FðU 0 ; a; qÞ > lqc À u20 þ lq0cÀ1 q P 0
 2
u0

l

þ qc0À1

1=ðcÀ1Þ 

c þ 1 1=ðcÀ1Þ
¼
qmax ðU 0 ; aÞ > u2 ðU 0 ; aÞ:
2
ð3:16Þ

Since

FðU 0 ; a; qÞ > 0;

4.1. Well-balanced numerical scheme
Given a time step Dt > 0 and a spacial mesh size Dx. Set

xj ¼ jDx; j 2 Z;

t n ¼ nDt; n 2 N;

k¼

Dt
;
Dx

2

d FðU 0 ; a; qÞ=dq2 > 0;

for q > u2 ðU 0 ; aÞ and q2 > u2 ðU 0 ; aÞ, the Newton–Raphson method
starting at q2 will generate a monotone decreasing sequence that
converges to u2 ðU 0 ; aÞ.
We can summarize the above argument in the following
algorithm which describes the choice for a starting point in the
Newton–Raphson method, applying to calculate the admissible
root of the nonlinear Eq. (3.12).
3.2.1. Algorithm of selecting admissible root
A pseudo-code for Newton–Raphson method selecting the
admissible root of Eq. (3.12) can be described as follows. We consider only for u0 > 0, since the case u0 6 0 can be treated similarly.
Algorithm 1. q ¼ SelectingRootðU 0 ; aÞ

while jFðU 0 ; a; qÞj < 1e À 12

 À
Á2
cÀ1

À a0 ua0 q0 q12 ;
¼ lcqcÀ1 À u20 þ lq0

dFðU 0 ;a;qÞ
dq

FðU 0 ;a;qÞ
q ¼ q À dFðU
0 ;a;qÞ=dq



À
Á2
cÀ1
FðU 0 ; a; qÞ ¼ lqc À u20 þ lq0
q þ a0 ua0 q0 q1 ;
end

ð4:2Þ

In the scheme (4.2), the states

U njþ1;À

¼ ðq; qu; qeÞnjþ1;À ;

U njÀ1;þ ¼ ðq; qu; qeÞnjÀ1;þ

are defined as follows. First, observe that the entropy is constant

across each stationary jump, we compute qnjþ1;À ; unjþ1;À from the
equations

anjþ1 qnjþ1 unjþ1 ¼ anj qnjþ1;À unjþ1;À ;
¼

ðunjþ1;À Þ2
2

ðunjþ1 Þ2
2

and we compute q

anjÀ1 qnjÀ1 unjÀ1 ¼ anj qnjÀ1;þ unjÀ1;þ ;
¼

ðunjÀ1;þ Þ2
2

þ hðqnjþ1 Þ

þ hðqnjþ1;À Þ;

n
n
jÀ1;þ ; ujÀ1;þ

ð4:3Þ

from the equations

ðunjÀ1 Þ2
2

þ hðqnjÀ1 Þ

þ hðqnjÀ1;þ Þ:

ð4:4Þ

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–Friedrichs numerical flux:

gðU; VÞ ¼
If k1 ðU 0 Þ P 0
- Start at q1 : q ¼ q1
else
- Starting at q2 : q ¼ q2
end


À
Á2
cÀ1
FðU 0 ; a; qÞ ¼ lqc À u20 þ lq0
q þ a0 ua0 q0 q1 ;

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



U nþ1
¼ U nj À k gðU nj ; U njþ1;À Þ À gðU njÀ1;þ ; U nj Þ ;
j

for

q P q2 :¼

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

1
1
ðf ðUÞ þ f ðVÞÞ À ðV À UÞ:
2
2k

ð4:5Þ

4.2. Computing correctors
As indicated above, the selection of the states U nj;Æ involve the

Monotone Criterion. So, when a rarefaction wave started at
U L 2 G2 approaches and reaches the resonant surface Rþ at a state
U 1 , it can be followed up by a stationary contact that is attached to
it at U 1 and jumps into U 2 2 G1 . The attaching condition means that
the characteristic speed k1 ðU 1 Þ at the end of the rarefaction fan
coincides with the discontinuity speed k0 ðU 1 ; U 2 Þ ¼ 0.
Let U nj be an approximation of such an above mentioned state
U 1 of a rarefaction wave from U L 2 G2 to U 1 2 Rþ . We need to control the distance between U nj and the resonant surface Rþ using
k1 ðU nj Þ. More precisely, whenever the following condition holds
n

k1 ðU nj Þ < Àdj ;
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. Moreover, it preserves the positivity of the density and possesses the
numerical minimum entropy principle, see [23]. As most existing
schemes, its original version may fail to approximate solutions when
data belong to both sides of the resonant surface. In particular, the
scheme may not work when a rarefaction wave is attached by a sta-

n
dj

where
> 0 represents a computing corrector, we will take the root
u1 ðU nj ; ajþ1 Þ to jump to U njþ1;À 2 G1 . Note that in this case U nj may belong to G1 or G2 , or Rþ . In the following we suggest two computing
correctors:
(I) A mesh-size dependent corrector:



n
dj ¼ Dxmax jki ðU nj Þj jqnjþ1 À qnj j þ junjþ1 À unj j þ jpnjþ1 À pnj j ;
i¼1;2;3

ð4:6Þ


134

M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139

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

¼

maxi¼1;2;3 jki ðU nj Þj 
pffiffiffi
k

n
jþ1

jq

n
jj

Àq þ

junjþ1

À

unj j

þ

jpnjþ1

together with the corresponding right-eigenvectors:

À

pnj j



0

;

ð4:7Þ
where k is the number of iterations.

1

1
0
1
q
ffiffiffiffiffiffiffiffiffiffiffi
p
ffiffiffiffiffiffiffiffiffiffiffi
p
B
C
B
C
B
C
r 0 :¼ @ Àp0 ðqÞ A r 1 :¼ @ À p0 ðqÞ A r 2 :¼ @ p0 ðqÞ A:
ap0 ðqÞ
0
0
au À u

qu

0

q

Set

The way to take into account one of the above computing correctors can be described in the following algorithm.

pffiffiffiffiffiffi cÀ1
CÆ : u ¼ Æ jc q 2 :
We can see that

4.2.1. Algorithm of computing admissible root
A pseudo-code for Newton–Raphson method selecting the
admissible root of the Eq. (3.12) can be described as follows. We
consider only for u0 > 0, since the case u0 6 0 can be treated
similarly.

k1 ¼ k0

on Cþ ;

k2 ¼ k0

on CÀ :

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

Algorithm 2. q ¼ ComputingCorrectorðU 0 ; a; dÞ

&
U 0 ðxÞ ¼

If k1 ðU 0 Þ P Àd
– Start at q1 : q ¼ q1
else
– Starting at q2 : q ¼ q2

end


À
Á2
cÀ1
FðU 0 ; a; qÞ ¼ lqc À u20 þ lq0
q þ a0 ua0 q0 q1 ;

ð5:2Þ

U 1 ¼ ð1:3783; 1:2616; 1Þ

while jFðU 0 ; a; qÞj < 1e À 12

 À
cÀ1
a u q Á2
¼ lcqcÀ1 À u20 þ lq0
À 0 a0 0 q12 ;

U 2 ¼ ð0:97819; 1:6161; 1:1Þ;
U 3 ¼ ð1:2304; 1:3387; 1:1Þ:

FðU 0 ;a;qÞ
dFðU 0 ;a;qÞ=dq



À

Á2
cÀ1
FðU 0 ; a; qÞ ¼ lqc À u20 þ lq0
q þ a0 ua0 q0 q1 ;

ð5:3Þ

The exact solution is a rarefaction wave from U L to U 1 2 Cþ , followed by a stationary wave from U 1 to U 2 , followed by a 1-shock
from U 2 to U 3 , and then arrives at U R by a 3-shock.
Without a corrector, the well-balanced method with underlying
Lax-Friedrichs scheme does not give a good approximation to the
exact solution, see Fig. 1.

end

5. Test cases

5.2. Test 2

5.1. Test 1
For Tests 1–3 below, the solution is evaluated for x 2 ½À1; 1Š
with the mesh sizes of 1000 points and 3000 points, and at the
time t ¼ 0:2. We take

C:F:L ¼ 0:5:
The following test is devoted to an isentropic ideal gas, where
the pressure is given by

p ¼ jqc ;

x < 0;

where G1 is the domain where k1 ðUÞ > 0, and G2 is the domain
where k1 ðUÞ < 0 and k2 ðUÞ > 0. In this test, the Riemann data are taken on the opposite sides of the resonance curve in the ðq; uÞ-plane.
Set

dFðU 0 ;a;qÞ
dq

q¼ qÀ

U L ¼ ðqL ; uL ; aL Þ ¼ ð3; 0:2; 1Þ 2 G2 ;

U R ¼ ðqR ; uR ; aR Þ ¼ ð1:4; 1:5; 1:1Þ 2 G1 ; x > 0;

In this test, we use the same data as in Test 1, and we equip our
well-balanced scheme by the modified version of computing
corrector I in (4.6) by neglecting the term of the pressure. The
underlying Lax-Friedrichs scheme is chosen. This tests shows that
our method provides us with a good approximations to the exact
solution with 1000 and 3000 mesh points for the interval ½À1; 1Š,
see Fig. 2.

c > 1; j > 0

and in the sequel, for simplicity we take j ¼ 1. The governing equations of the model of the isentropic fluid in a nozzle with variable
cross-section are given by

@ t ðaqÞ þ @ x ðaquÞ ¼ 0;
@ t ðaquÞ þ @ x ðaðqu2 þ pÞÞ ¼ p@ x a;

x 2 R; t > 0;

ð5:1Þ

Let us recall some basic properties of the model (5.1). The reader is referred to [28] for more details. Taking the variable
U ¼ ðq; u; aÞ, we can re-write the system (5.1),(1.2) in the form

@ t U þ AðUÞ @ x U ¼ 0;
where

0

u
q qu=a 1
@
AðUÞ ¼ h0ðqÞ u
0 A;
0
0
0

hðqÞ ¼

jc cÀ1
q :
cÀ1

The matrix AðUÞ admits the following three eigenvalues

k0 :¼ 0;

k1 :¼ u À

pffiffiffiffiffiffiffiffiffiffiffi
p0ðqÞ;

k2 :¼ u þ

pffiffiffiffiffiffiffiffiffiffiffi
p0ðqÞ;

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


135

M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139

Fig. 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.

R1 ðU L ; U 1 Þ ! W 0 ðU 1 ; U 2 Þ ! S1 ðU 2 ; U 3 Þ ! W 2 ðU 3 ; U 4 Þ ! S3 ðU 4 ; U R Þ;
ð5:4Þ
where R1 ðU L ; U 1 Þ stands for a 1-rarefaction wave from U L to
U 1 ; W 0 ðU 1 ; U 2 Þ stands for a stationary contact from U 1 to
U 2 ; S1 ðU 2 ; U 3 Þ stands for a 1-shock from U 2 to U 3 ; W 2 ðU 3 ; U 4 Þ stands
for a 2-contact from U 3 to U 4 , and S3 ðU 4 ; U R Þ stands for a 3-shock
from U 4 to U R . The computing strategy of the states
U i ; i ¼ 1; 2; 3; 4 can be shown bellow.

Setting

mðSÞ ¼ c1=2 ðc À 1Þ1=2c exp

S À SÃ
;
2cC v

nðSÞ ¼

2mðSÞ
;
cÀ1

w¼

cÀ1
;
2c
ð5:5Þ

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

5.3. Test 3
This test is devoted to a nonisentropic fluid, where we take

c ¼ 1:4; C v ¼ 1; SÃ ¼ 1:
First, let us describe a way of computing exact solutions where
data belong to both sides of the resonant surface and that a rarefaction wave in one side is attached by a stationary wave that jumps
into the other side.

we can rewrite the resonant surface Rþ as

Rþ : u ¼ mðSÞpw ;

ð5:6Þ

and the 1-wave rarefaction curve R1 ðU 0 Þ as (see [38])

R1 ðU 0 Þ : u ¼ u0 À nðS0 Þðpw À pw
0 Þ;

p 6 p0 :

ð5:7Þ

The state U 1 satisfies the equation
w
w
u1 ¼ mðSL Þpw
1 ¼ uL À nðSL Þðp1 À pL Þ

ð5:8Þ

which gives
5.4. Computing the typical Riemann solutions
As indicated in [38], we can construct Riemann solutions by
projecting all the wave curves in the ðp; uÞ-plan. In particular, a Riemann solution can begins with a 1-rarefaction wave from
U L ¼ ðpL ; uL ; SL ; aL Þ 2 G2 and lasts until the rarefaction touches the
resonant surface Rþ at a state U 1 ¼ ðp1 ; u1 ; SL ; aL Þ since the entropy

is a Riemann invariant. At U 1 , the characteristic speed k1 ðU 1 Þ ¼ 0
and the solution can use a stationary contact to jump to a state
U 2 ¼ ðp2 ; u2 ; SL ; aR Þ 2 G1 . The intersection in the ðp; uÞ-plan of the
forward 1-wave curve W 1 ðU 2 Þ and the backward 3-wave curve
W 3 ðU R Þ consists of one state U 3 . The Riemann solution is thus continued by a 1-wave from U 2 to U 3 . This wave is a 1-shock if p2 < p3
and a 1-rarefaction wave otherwise. In the computing strategy below we will choose a shock. This 1-wave is followed by a 2-contact
from U 3 to U 4 ¼ ðp3 ; u3 ; S4 ; aR Þ.
The solution then arrives at U R by a 3-wave from U 4 . This wave
is a 3-shock if p3 > pR and a 3-rarefaction wave otherwise. In the
sequel we will choose a shock. Thus, the solution has the form
Fig. 3

p1 ¼



1=w
uL þ nðSL Þpw
L
:
mðSL Þ þ nðSL Þ

ð5:9Þ

So, the state U 1 ¼ ðp1 ; u1 ; SL ; aL Þ is determined by (5.9) and then
(5.8).
Next, to evaluate U 2 , we rewrite the state U 1 into the form
U 1 ¼ ðq1 ; u1 ; SL ; aL Þ with the q-component instead of the p-component using the equation of state

q ¼ qðp; SÞ ¼




1=c
p
SÃ À S
:
exp
Cv
cÀ1

The state U 2 ¼ ðq2 ; u2 ; SL ; aR Þ is obtained by a stationary contact
from U 1 . As indicated earlier, we calculate the q-component of U 2
using the Newton–Raphson method. The u-component of U 2 is followed immediately.
Calculations for the Riemann solution continue with the determination of U 3 as follows. We rewrite U 2 in the form
U 2 ¼ ðp2 ; u2 ; SL ; aR Þ. In the ðp; uÞ-plane, the intersection of S 1 ðU 2 Þ
and S 3 ðU R Þ determines U 3 . It is necessary that uR < u2 . As shown
in [38], S 1 ðU 0 Þ and S 3 ðU 0 Þ are given by


136

M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139

Table 1
States that separate the elementary waves of the exact Riemann solution in Test 3, see
Fig. 3.

UL

U1
U2
U3
U4
UR

q

u

p

a

5
2.7766
1.6697
2.0779
1.8047
1

0.5
1.3306
1.8438
1.5738
1.5738
0.8

8
3.5111

1.7227
2.3427
2.3427
1

1
1
1.2
1.2
1.2
1.2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv 0
S 1 ðU 0 Þ : u ¼ u0 À ðp À p0 Þ
; p P p0 ;
p þ lp0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv 0
; p P p0 ;
S 3 ðU 0 Þ : u ¼ u0 þ ðp À p0 Þ
p þ lp0

l¼

cÀ1
:
cþ1

It is easy to see that the function f1 ðpÞ in (5.12) is strictly

increasing and strictly concave. So, the root p ¼ p3 of the nonlinear
Eq. (5.12) can also be calculated using the Newton–Raphson method, where

f10 ðpÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv R
ðp þ lpR ÞÀ1=2
¼
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv R
þ
ð1 þ lÞpR ðp þ lpR ÞÀ3=2
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv 2
ðp þ lp2 ÞÀ1=2
þ
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv 2
þ
ð1 þ lÞp2 ðp þ lp2 ÞÀ3=2 > 0:
2

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

U 0 ðxÞ ¼

&

ð5:10Þ
Thus, the state U 3 satisfies the equations

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv 2
u ¼ u2 À ðp À p2 Þ
p þ lp2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv R
;
¼ uR þ ðp À pR Þ
p þ lpR

p P maxfp2 ; pR g;

ð5:11Þ

where v ¼ 1=q. It is derived from (5.11) that the p-component of U 3
is the root of the nonlinear equation

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 À lÞv R
ð1 À lÞv 2
þ ðp À p2 Þ
þ uR À u2 ¼ 0:
f1 ðpÞ ¼ ðp À pR Þ

p þ lpR
p þ lp2
ð5:12Þ

U L ¼ ðqL ; uL ; pL ; aL Þ ¼ ð5; 0:5; 8; 1Þ 2 G2 ;

x < 0;

U R ¼ ðqR ; uR ; pR ; aR Þ 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 U L 2 G2 and U R 2 G1 . The solution has
the form (5.4), where the states that separate elementary waves of
the Riemann solution are given in Table 1.
Our well-balanced scheme ((4.1)–(4.5)) equipped by either Corrector I in (4.6) or Corrector II in (4.7) gives good approximations as
indicated in Fig. 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

Fig. 4. Test 3. Our well-balanced method equipped by a computing corrector gives a good approximation to the exact pressure.


137

M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139
Table 2
States that separate the elementary waves of the exact Riemann solution in Test 4, see

Fig. 5.
n

q

u

p

a

UL
U1
U2
U3
U4
U5
UR

1.3
1.872903
3.775791
2.969906
0.533582
1
2.363115

2
1.775738
0.880818

1.250641
3.067818
3.067818
3.675948

1
1.66725
4.643562
3.318027
0.3
0.3
1

1
0.78177
0.78177
0.7
0.7
0.7
0.7

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.

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 U L

to U 1 , followed by a 1-shock with zero speed from U 1 to U 2 , then
followed by another stationary wave from U 2 to U 3 . It is attached
by a rarefaction wave from U 3 to U 4 , and it contiues with a 2-contact discontinuity from U 4 to U 5 , and finally it reaches U R by a
3-shock. See Fig. 5, where the states U 1 ; U 2 , and U 3 are distributed
along the t-axis in the ðx; tÞ-plane.
Our scheme using Corrector (II) in (4.7) computes the approximate solution at the time t ¼ 0:2s over the interval ½0; 2Š with 6000
mesh points. As in [33], the initial discontinuity is located at
x ¼ 0:8. The density and velocity of the exact and approximate
solutions are displayed in Fig. 6. Fig. 6 shows that the scheme
can provides us with a good approximation to the exact solution.

Table 3
States that separate the elementary waves of the exact Riemann solution in Test 5, see
Fig. 7.

UL
U1
U2
U3
U4
U5
UR

q

u

p

a

2.363115
1
0.533582
2.969906
3.775791
1.872903
1.3

À3.675948
À3.067818
À3.067818
À1.250641
À0.880818
À1.775738
À2

1
0.3
0.3
3.318027
4.643562
1.66725
1

0.7
0.7
0.7
0.7
0.78177

0.78177
1

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.

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. Precisely, these states are given in Table 3.
The exact Riemann solution starts by a 1-rarefaction wave from
U L to U 1 , followed by a 2-contact discontinuity from U 1 to U 2 , then
followed by a rarefaction wave from U 2 to U 3 . It is attached by a
stationary wave from U 3 to U 4 , followed by a 1-shock wave with
zero speed from U 4 to U 5 , and finally it reaches U R by another stationary wave. See Fig. 7, where the states U 3 ; U 4 , and U 5 are distributed along the t-axis in the ðx; tÞ-plane.

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
Corrector (II) in (4.7) over ½0; 2Š at the time t ¼ 0:2s.


138

M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139

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; 2Š 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.

UL
U1
U2
U3
U4
U5
UR

q

u

p

a

1.4
2.077419
0.3
0.258647
0.200487
0.232799
0.269081

À2
À2.591083

À2.591083
À3.379023
À4.359267
À3.837976
À3.273959

2
3.5
3.5
2.84374
1.98703
2.449388
3

1
1
1
0.889412
0.889412
0.87
0.87

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.

Our scheme using Corrector (II) in (4.7) computes the approximate solution at the time t ¼ 0:12s over the interval ½0; 2Š with
6000 mesh points. As in [33], the initial discontinuity is located
at x ¼ 0:8. The density and velocity of the exact and approximate
solutions are displayed in Fig. 8, where we display the solutions
over the interval ½0; 1:2Š 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 U L to
U 1 , followed by a 2-contact discontinuity from U 1 to U 2 , then
followed by a stationary wave from U 2 to U 3 . It continues with a
1-shock with zero speed from U 3 to U 4 , followed by another stationary wave from U 4 to U 5 , and finally it reaches U R by a 3-rarefaction wave. See Fig. 9, where the states U 3 ; U 4 , and U 5 are
distributed along the t-axis in the ðx; tÞ-plane.
Our scheme using Corrector (II) in (4.7) computes the approximate solution at the time t ¼ 0:15s over the interval ½0; 2Š with
6000 mesh points. As in [33], the initial discontinuity is located
at x ¼ 0:8. The density and velocity of the exact and approximate
solutions are displayed in Fig. 10 in the interval ½0; 1:5Š. Fig. 10 also

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
Corrector (II) in (4.7) over ½0; 2Š at the time t ¼ 0:15s.


M.D. Thanh, D. Kröner / Computers & Fluids 66 (2012) 130–139

indicates that our scheme can provide a reasonable approximation
to the exact solution.

6. Conclusions
Most existing schemes for nonconservative systems or nonstrictly hyperbolic systems can approximate the exact solutions only in
strictly hyperbolic domains. This work gives a way to treat numerically the nonconservative terms of the model of a fluid in a nozzle

with variable cross-section in the resonant regime where data
belong to both sides of the resonant surface. We introduce two
types of computing correctors to ‘‘navigate’’ the scheme to take
the right state. Tests show that our well-balanced method
equipped by one of these computing correctors gives good approximations. Questions on a general approach and higher-order
schemes are open for further study.
Acknowledgments
The authors are grateful to the reviewers for their very constructive comments and helpful suggestions.
This research is funded by Viet Nam National Foundation for
Science and Technology Development (NAFOSTED) under Grant
No. 101.02-2011.36.
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