DSpace at VNU: On traveling waves in viscous-capillary Euler equations with thermal conductivity

Given a shock wave, we can obtain exactly the sign of the real part of each eigenvalue of the Jacobian matrix of the corresponding system of ODEs at the two equilibria associated with th

On traveling waves in viscous-capillary Euler equations

with thermal conductivity

a

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

b

Faculty of Applied Science, University of Technology, 268 Ly Thuong Kiet str., District 10, Ho Chi Minh City, Viet Nam

a r t i c l e i n f o

Keywords:

Gas dynamics equations

Traveling wave

Shock

Viscosity

Capillarity

Thermal conductivity

Equilibria

Stability

a b s t r a c t

This work establishes the stability of the equilibrium states corresponding to traveling waves in viscous-capillary Euler equations when a standard thermal conductivity coefficient is present Due to the presence of the heat conduction, the associated system

of ordinary differential equations is much more involved and complicated Given a shock wave, we can obtain exactly the sign of the real part of each eigenvalue of the Jacobian matrix of the corresponding system of ODEs at the two equilibria associated with the left-hand and right-hand states of the shock It turns out that one equilibrium point is asymptotically stable, why the other point is unstable and admits eigenvalues whose real parts have opposite signs Suitable approximate connections between the unstable and stable equilibria are obtained by numerical tests for various ranges of thermal conductivity: low, medium, and high values Moreover, numerical tests also suggest that a trajectory could leave the unstable equilibrium point and enters the attraction domain of the asymptotically stable equilibrium point This work therefore may motivate for further study

on the existence of the traveling waves of the viscous-capillary gas dynamics equations with the presence of heat conduction

Ó 2014 Elsevier Inc All rights reserved

1 Introduction

Heat conduction causes a tough obstacle for studying traveling waves in gas dynamics equations Therefore, the existence

of traveling waves for gas dynamics equations with thermal conductivity is a very interesting and challenging problem This work concerns with the traveling waves of the following model of fluid dynamics equations with viscosity, capillarity, and heat conduction

vt ux¼ 0;

utþ px¼ k

vux

x

 ðl vxÞxxþ lv

2 v2 x

x;

Etþ ðupÞx¼ k

vuux

x

þ lv

2 uv2

x uðl vxÞx

xþðluxvxÞxþ j

vTx

x

ð1:1Þ

http://dx.doi.org/10.1016/j.amc.2014.02.004

0096-3003/Ó 2014 Elsevier Inc All rights reserved.

⇑Corresponding author.

E-mail addresses: mdthanh@hcmiu.edu.vn (M.D Thanh), nguyenhuuhiep@hcmut.edu.vn (N.H Hiep).

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

for x 2 R and t > 0 Here,v;S; p;e;T denote the specific volume, entropy, pressure, internal energy, temperature, respec-tively; u is the velocity, and

E ¼eþu

2

2þ

l

2v2

is the total energy The non-negative quantities k;l;jrepresent the viscosity, capillarity, and the heat conduction, respec-tively Observe that the Lagrangian coordinates are chosen so that the calculations are simple only, since similar results hold for the Eulerian coordinates

It has been known that traveling waves can be used to justify an admissibility criterion for shock waves of different types, such as the Lax shocks[13]and nonclassical shocks[11] However, the study of traveling waves has been restricted mostly to the case where additional terms are merely viscosity and capillarity Heat conduction is commonly ignored The obstacle for the study of traveling waves when heat conduction is present is that this quantity causes tough inconvenience to establish a system of ordinary differential equations whose equilibria are used to characterize the traveling waves

Recently, the system of ordinary differential equations associated with traveling waves of(1.1) was derived by[21], where the equilibrium points of the system were shown to correspond to the left-hand and right-hand states of the given shock In this work, we first establish the stability of these corresponding equilibrium points Precisely, by investigating the sign of the real part of each eigenvalue of the Jacobian matrix of the resulted system of ODEs at the two equilibrium points, we can show that one equilibrium point is asymptotically stable, and the other equilibrium point is unstable and

it furthermore admits eigenvalues whose real parts are of opposite signs Thus, existence of a traveling wave can be expected and could be a topic for further study For example, whenever an unstable trajectory leaves the unstable equilibrium point at

1 and enters the attraction domain of the asymptotically stable equilibrium point, it will converge to the asymptotically stable equilibrium point at 1, and so a traveling wave is obtained Then, we present numerical tests, which all show that the trajectory starting very closed to the unstable equilibrium point converges to the asymptotically stable equilibrium point Traveling waves have attracted attention of many authors An early work on the related shock layers of the gas dynamics equations with viscosity and heat conduction effects (with zero capillarity) was presented in[10] Traveling waves for dif-fusive-dispersive scalar equations were earlier studied in[7,12] Traveling waves of the hyperbolic-elliptic model of phase transition dynamics were considered in[16,17,8,9,15] The existence of traveling waves associated with Lax shocks for vis-cous-capillary models was considered in our recent works[19–23] These works developed the method of estimating attrac-tion domain of the asymptotically stable equilibrium point to establish the existence of traveling waves The existence of traveling waves corresponding to nonclassical shocks for viscous-capillary models was considered in[11,3,4,2,5,6,1] See also the references therein

The organization of this paper is as follows Section2provides basic concepts and properties of the fluid dynamics equa-tions and the autonomous system of ordinary differential equaequa-tions for a given traveling wave Section3is devoted to the stability of the resulted equilibria of the corresponding system of ODEs In Section4we present several numerical tests, where approximate traveling waves are computed for both cases of positive and negative shock speeds, with large and small thermal conductivity In Section5we will provide some conclusions and discussions

2 Preliminaries

2.1 Shock waves and Lax shock inequalities

If one lets the viscosity, the capillarity and the thermal conductivity coefficients in(1.1)tend to zero, one obtains the fluid dynamics equations in the Lagrangian coordinates

vt ux¼ 0;

utþ px¼ 0;

Etþ ðupÞx¼ 0; x 2 R; t > 0:

ð2:1Þ

As well-known, the system(2.1)can be written in terms of the variable U ¼ ðv;u; SÞ by

vt ux¼ 0;

utþ pvðv;SÞvxþ pSðv;SÞSx¼ 0;

St¼ 0:

The Jacobian matrix of the last system is given by

A ¼

pv 0 pS

0

B

B

1 C

C;

which admits three distinct real eigenvalues

k1¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pvðv;SÞ

p

<k2¼ 0 < k3¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pvðv;SÞ

p

;

so that the system(2.1)is strictly hyperbolic, whenever pvðv;SÞ < 0 The value kiis often referred to as the characteristic speed of the ith characteristic field, i ¼ 1; 2; 3

Let Uand Uþbe given constant states A shock wave of(2.1)connecting the left-hand state Uand the right-hand state

Uþis a weak solution U of the form

Uðx; tÞ ¼ U; if x < st;

Uþ; if x > st;



ð2:2Þ

with a constant shock speed s The shock speed s can be determined via the following Rankine–Hugoniot relations for this shock

sðvþvÞ þ ðuþ uÞ ¼ 0;

 sðuþ uÞ þ pþ p¼ 0;

eþeþpþþ p

2 ðvþvÞ ¼ 0;

ð2:3Þ

that is

s ¼ sðU;UþÞ ¼ 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pþ p

vþv

r

whenever

pþ p

vþv

P0:

It has been known that a negative shock speed corresponds to the first characteristic field, and a positive shock speed cor-responds to the third characteristic field of(2.1)

A shock wave is admissible under a certain admissibility criterion The most standard admissibility criterion is the Lax shock inequalities which requires the shock speed s ¼ sðU;UþÞ to satisfy

2.2 System of ODEs for traveling waves

A traveling wave of (1.1) connecting a left-hand state, denoted by U, and a right-hand state, denoted by Uþ, is a smooth solution of (1.1) of the form U ¼ UðyÞ; y ¼ x  st, where s is a constant, and satisfies the following boundary conditions

limy!1UðyÞ ¼ U

limy!1

d

dyUðyÞ ¼ limy!1

d2

dy2UðyÞ ¼ 0:

ð2:6Þ

The derivation of the systems of ordinary differential equations for traveling waves of(1.1)written in the ðv;u; SÞ variable was obtained in[24] Here, we present another form of systems of ODEs for traveling waves of(1.1)as follows Arguing as in [24], we have the following differential equations for traveling waves of(1.1)

u0¼ sv0;

ðl v0Þ0¼ sk

v v0þ

lv

2 ðv0Þ2 ðp  pÞ  s2ðvvÞ;

s

2lðv0Þ2¼ sðeeÞ þ spðvvÞ s

3

2ðvvÞ2þj

vT

0

:

ð2:7Þ

Let us choose the temperature variable as the unknown and by re-writing the second equation of(2.7)in terms of the tem-perature, we obtain from(2.7)

ðl v0Þ0¼ sk

v v0þ

lv

2 ðv0Þ2 ðp  pÞ  s2ðvvÞ;

T0

¼sv

j

1

2lðv0Þ2 ðeeÞ þ pðvvÞ s

2

2ðvvÞ2

:

ð2:8Þ

The equations(2.8)can be written as the following system of first-order ordinary differential equations

v0¼w

l;

w0¼ sk

l vw þ

lvðv;Sðv;TÞÞ

2l2 w2 p  p þ s2ðvvÞ

;

T0¼sv

j

1

2lðv0Þ2 ðeeÞ þ pðvvÞ s

2

2ðvvÞ2

;

ð2:9Þ

in terms of the variable ðv;w; TÞ Observe that the system(2.9)is common for a general fluid Equilibrium points of(2.9) satisfy

w ¼ 0;

p  pþ s2ðvvÞ ¼ 0;

ðeeÞ þ pðvvÞ s

2

2ðvvÞ2¼ 0:

ð2:10Þ

Moreover, integrating the equation

u0¼ sv0

from 1 to 1 gives us

From(2.10) and (2.11), we obtain the Rankine-Hugoniot relations(2.3) Thus, U¼ ðv;u;pÞ can be connected with each other by a shock wave of(2.1)

The system of ODEs(2.9)has the advantage to be common for every fluid However, investigating the stability of its equi-librium points(2.10)would need other forms which rely on the specific equation of state of the fluid In[24], the system of ODEs for ideal fluids, stiffened gas EOS, and van der Waals fluids were obtained as follows

2.3 System of ODEs for ideal fluids

An ideal fluid is governed by the equation of state

wherec>1 is the adiabatic constant Using the thermodynamic identity, we can write

p ¼ pðv;SÞ ¼ ðc 1Þv cexp S S

c

ð2:13Þ

for some constant S, where c is the specific heat coefficient at constant volume

Any traveling wave of(1.1)satisfies the relation(2.11), and the following system of ordinary differential equations in terms of the variable ðv;w; SÞ

v0¼w

l;

w0¼ sk

l vw þ

lv

2l2w2

 pð v;SÞ  pðv;SÞ þ s2ðvvÞ

;

S0

¼ðc 1Þc

c2sðc 1Þ

jpðv;SÞ

1

2lw

2 eðv;SÞ eþ pðvvÞ s

2

2ðvvÞ2

:

ð2:14Þ

2.4 System of ODEs for a stiffened gas EOS

A stiffened gas EOS is given by

wherec>1 ande;pare parameters Another form of stiffened gas EOS is

p ¼ pðv;SÞ ¼ ðc 1Þv cexp S  S

c

for some constant S

An arbitrary traveling wave of(1.1)for a stiffened gas satisfies the relation(2.11), and the following system of ordinary differential equations

v0¼w

l;

w0¼ sk

l vw þ

lv

2l2w2 pð v;SÞ  pðv;SÞ þ s2ðvvÞ

;

S0

¼ðc 1Þc

c2sðc 1Þ

jðpðv;SÞ þ p1Þ

1

2lw

2

 eðv;SÞ eþ pðvvÞ s

2

2ðvvÞ2

:

ð2:17Þ

2.5 System of ODEs for van der Waals fluids

A van der Waals fluid is defined by the equation of state

p ¼ RT

v b

a

where a > 0; b > 0 and R > 0 are constants A van der Waals fluid can be fully determined by considering the following Helm-holtz free energy function

Fðv;TÞ ¼ RT 1 þ ln ðv bÞT3=2

c

!

a

where c > 0 is a parameter, see[18]for example The Helmholtz free energy determines the specific entropy as

S ¼ @TFðv;TÞ ¼ R ln ðv bÞT3=2

c

!

þ5 2

! :

Thus, one can resolve the specific entropy from the last equation by

T ¼ Tðv;SÞ ¼ d

ðv bÞ2=3exp

2S 3R

5 3

; d ¼ c2=3:

The last equality and the equation of state(2.18)imply that the pressure can be expressed by

p ¼ pðv;SÞ ¼ Rd

ðv bÞ5=3exp

2S 3R

5 3

 a

Every traveling wave of(1.1)for a fluid of van der Waals(2.18)satisfies the relation(2.11), and the following system of ordinary differential equations

v0¼w

l;

w0¼ sk

l vw þ

lv

2l2w2

 pð v;SÞ  pðv;SÞ þ s2ðvvÞ

;

S0

lðv bÞþ

cvsv

jTðv;SÞ

1

2lw

2 eðv;SÞ eþ pðvvÞ s

2

2ðvvÞ2

:

ð2:21Þ

3 Stability of the equilibria of traveling waves

In this section, let us investigate the stability of the two equilibria of(2.9)under the condition(2.10) As mentioned in Section2, the corresponding states Ucan be connected by a shock wave of(2.1) And we expect that the stability property

of these equilibria could yield the existence of a traveling wave of(1.1)connecting both states of the shock

3.1 Stability of the equilibria of traveling waves for ideal fluids

Theorem 3.1 (Stability of equilibria for ideal fluids) Consider a shock of (2.1) of the form(2.2) satisfying the Lax shock inequalities(2.5)for an ideal fluid

(i) For s > 0: The equilibrium point ðvþ;0; SþÞ of(2.14)is asymptotically stable; the point ðv;0; SÞ is unstable, where the Jacobian matrix of(2.14)at ðv;0; SÞ has one positive eigenvalue, and the other two eigenvalues are either real and neg-ative, or complex and have negative real parts

(ii) For s < 0: The equilibrium point ðv;0; SÞ of(2.14)is asymptotically stable in the negative direction (y ! 1); the point

ðvþ;0; SþÞ is unstable, where the Jacobian matrix of(2.14)at ðvþ;0; SþÞ has one negative eigenvalue, and the other two eigenvalues are either real and positive, or complex and have positive real parts

Proof We need only prove for the case (i), since the argument for the case (ii) is similar A straightforward calculation shows that the Jacobian matrix of the system(2.14)at V¼ ðv;0; SÞ is given by

A

¼

pvðv;SÞ  s2  sk

lv p

c

0 ðclv1Þc csv

j

0

B

B

@

1 C C

where p¼ pðv;SÞ The characteristic equation of the matrix Acan be written in the form

X3þ a

2X2þ a

1X þ a

0¼ 0;

where

a

0 ¼1

lðs

2þ pvðv;SÞÞcs

j v;

a

1 ¼1

2

þ pvðv;SÞ þðc 1Þp

v

þcs

2

k

j

;

a

2 ¼ sk

l vþcs

j v:

ð3:2Þ

First, consider the point ðvþ;0; SþÞ The Lax shock inequalities imply that

s2>pvðvþ;SþÞ:

This yields

aþ

0 ¼1

lðs

2þ pvðvþ;SþÞÞcs

j vþ>0;

aþ

1 ¼1

l s

2þ pvðvþ;SþÞ þðc 1Þpþ

vþ

þcs

2k

j

>0;

aþ

2 ¼lskvþþcs

j vþ>0:

So, all the coefficients of the characteristic equation

X3

þ aþ

2X2

þ aþ

1X þ aþ

are positive and its all real roots must therefore be negative It would be a contradiction if the contrary is assumed, since the left-hand side of(3.3)would be positive Thus, if all three roots of(3.3)are real, then they must be negative Otherwise,(3.3) must have exactly one real root, denoted by X1, and two complex conjugate roots, denoted by X2;X3 This real root r ¼ X1

must also be negative, as argued above We remain to show that the real parts of X2and X3are negative Indeed, one can verify easily that the factorization of the cubic polynomial

aX3

þ bX2þ cX þ d ¼ ðX  rÞðaX2þ ðb þ arÞX þ c þ br þ ar2Þ

gives the other two roots of the cubic equation by

ðb  raÞ  ffiffiffiffi

D

p

2

 4ac  2abr  3a2r2:

The two complex roots of the cubic equation(3.3)are given by

X2;3¼ðb þ rÞ 

ffiffiffiffi

D

p

2

where

b ¼ aþ

2; c ¼ aþ

1; d ¼ aþ

0:

Since X2;X3are complex, we haveD<0, and ReðX2Þ ¼ ReðX3Þ ¼ ðb þ rÞ=2 Set

f ðXÞ ¼ X3þ aþ

2X2þ aþ

1X þ aþ

0:

It is not difficult to check that

f ðbÞ ¼ aþ

2aþ

1þ aþ

0 <0; f ð0Þ ¼ aþ

0 >0:

This means that there is a real root in the interval ðb; 0Þ and since r < 0 is the unique real root of the cubic function f ðXÞ, it holds that

r > b:

Thus

ReðX2Þ ¼ ReðX3Þ ¼ ðb þ rÞ=2 < 0:

This establishes the stability of the point ðvþ;0; SþÞ

Next, consider the point ðv;0; SÞ The Lax shock inequalities imply that

s2<pvðv;SÞ:

This yields

a

0¼1

lðs

2

þ pvðv;SÞÞcs

j v<0; a

2 ¼ sk

l vþcs

j v>0:

It is derived from Viète’s theorem that the roots X1; X2and X3(real or complex) of the cubic equation

X3þ a

2X2þ a

1X þ a

0 ¼ 0

satisfy

X1þ X2þ X3¼ a

2<0;

X1X2X3¼ a

A cubic equation must have at least one real root The other two roots must be either both real or both complex and con-jugate If these two roots are also real, then the third inequality in (3.5)means that one root is positive, and the other two roots have the same sign Then, we deduce from the first inequality of(3.5)that one root is real and two roots are neg-ative Otherwise, let X1be the real root and let X2and X3be the two complex conjugate roots Then, the first inequality in (3.5)implies that ReðX2Þ ¼ ReðX3Þ < X1<0 This completes the proof ofTheorem 3.1 h

As seen byTheorem 3.1, it is possible that an unstable trajectory could leave for s > 0 the unstable equilibrium point

ðv;0; SÞ at y ¼ 1 and may enter the attraction domain of the asymptotically stable equilibrium point ðvþ;0; SþÞ If so, the unstable trajectory will then converge to the point ðvþ;0; SþÞ as y ! 1 This establishes a traveling wave connecting the corresponding points ðv;u;SÞ Similar observations can be made for the case s < 0

3.2 Equations for traveling waves for stiffened gas EOS

The following theorem establishes the stability of the equilibria of(2.17)for stiffened gas EOS

Theorem 3.2 (Stability of equilibria for stiffened gas EOS) Consider a shock of(2.1)of the form(2.2)satisfying the Lax shock inequalities(2.5)for an ideal fluid

(i) For s > 0: The equilibrium point ðvþ;0; SþÞ of(2.17)is asymptotically stable; the point ðv;0; SÞ is unstable, where the Jaco-bian matrix of(2.17)at ðv;0; SÞ has one positive eigenvalue, and the other two eigenvalues are either real and negative, or complex and have negative real parts

(ii) For s < 0: The equilibrium point ðv;0; SÞ of(2.17)is asymptotically stable in the negative direction (y ! 1); the point

ðvþ;0; SþÞ is unstable, where the Jacobian matrix of(2.17)at ðvþ;0; SþÞ has one negative eigenvalue, and the other two eigenvalues are either real and positive, or complex and have positive real parts

Proof It is not difficult to verify that the Jacobian matrix of the system(2.17)at V¼ ðv;0; SÞ is given by

B

¼

pvðv;SÞ  s2  sk

lv p þp 1

c

lv csv

j ;

0

B

@

1 C

where p¼ pðv;SÞ The characteristic equation of A

2 is given by

X3

þ bX2

þ bX þ b

b0 ¼1

lðs

2þ pvðv;SÞÞcs

j v;

b1 ¼1

2þ pvðv;SÞ þðc 1Þðpþ p1Þ

v

þcs

2k

j

;

b2 ¼ sk

l v

þcs

j v:

ð3:8Þ

Thus, the remaining of the proof can be made similarly as the one ofTheorem 3.1 h

3.3 Equations for traveling waves for van der Waals fluids

For a van der Waals fluid(2.18), the corresponding system of ODEs(2.21)may admit up to four equilibria Nonclassical shocks may appear in a van der Waals fluid A shock wave of this kind may violate the Lax shock inequalities For both kinds

of classical and nonclassical shocks, the shock speed is often compared with the characteristic speeds at the left-hand and right-hand states of the shock The reader is referred to the standard material[14]for the concept of classical and nonclas-sical shocks in hyperbolic systems of conservation laws So, in order to include both kinds of clasnonclas-sical and nonclasnonclas-sical shocks, we need to state the result on the stability of the equilibria of(2.21)in a more general form as in the following theorem

Theorem 3.3 (Stability of equilibria for van der Waals fluids) Consider a shock(2.2)of(2.1)for a van der Walls fluid with the shock speed s > 0, and the corresponding system of ODEs(2.21)of traveling waves of(1.1) If V0¼ ðv0;0; S0Þ is an equilibrium point of(2.21)resulted from the given shock, then the following conclusions hold

(i) If s2þ pvðv0;S0Þ > 0, then the equilibrium point ðv0;0; S0Þ of(2.21)is asymptotically stable

(ii) If s2þ pvðv0;S0Þ < 0, then the point ðv0;0; S0Þ is unstable, where the Jacobian matrix of(2.21)at ðv0;0; S0Þ has one positive eigenvalue, and the other two eigenvalues are either real and negative, or complex and have negative real parts Similar conclusions also hold for the case of negative shock speeds

Proof It is derived from(2.20)that

@vpðv;SÞ ¼ 5Rd

3ðv bÞ8=3exp

2S 3R

5 3

þ2a

v3;

@Spðv;SÞ ¼ 2d

3ðv bÞ5=3exp

2S 3R

5 3

:

ð3:9Þ

Since the equilibrium point V0is resulted from the shock between the left-hand and the right-hand states U, the state

U0¼ ðv0;u0¼ u sðv0vÞ; S0Þ, when considered as the right-hand state, satisfies the Rankine–Hugoniot relations(2.3)

So, a straightforward calculation gives the Jacobian matrix of the system(2.21)at the equilibrium point V0¼ ðv0;0; S0Þ by

C ¼

p0

vðv0;S0Þ  s2 sk

lv0 pSðv0;S0Þ

lðv0 bÞ  cv sv0

jTðv0 ;S 0 ÞeS

0

B

B

@

1 C C

It is not difficult to check that the characteristic equation of C can be written as

X3

þ c2X2

where

c0¼1lðpvðv0;S0Þ þ s2Þcvsv0

c1¼1

l pvðv0;S0Þ þ s2þcvs

2k

RpSðv0;S0Þ

v0 b

;

c2¼ sk

l v þ

cvsv0

ð3:12Þ

Consider the case (i) It is derived from(3.9)that pSðv;SÞ is always positive Thus, if s2þ pvðv0;S0Þ > 0, then all the coeffi-cients c0;c1and c2given by(3.12)of the characteristic polynomial(3.11)are positive The remaining proof of (i) and the proof of (ii) are therefore similar to the ones ofTheorem (3.1) h

4 Numerical tests

Since the point of the paper is to study traveling waves when a thermal conductivity coefficient is available, the following tests will be based on different ranges of thermal conductivity: low, medium, and large values Moreover, we also include the two cases of positive and negative shock speeds

4.1 Test 1: Traveling wave approximating a shock with positive speed

Let us consider numerical approximations of a traveling wave of(1.1)for an ideal fluid This traveling wave can be used to approximate a given shock wave between a left-hand state denoted by Uand a right-hand state denoted Uþwith a positive shock speed The parameters for this test are chosen by

c¼ 1:4;

S¼ 0:3;

c ¼ 1=2;

k¼ 1=5;

l¼ 5k;

j¼ 1=3;

ðv;u;S;pÞ ¼ ð2; 1; 3; 33:558937Þ;

ðvþ;uþ;Sþ;pþÞ ¼ ð3; 2:8317144; 2:9961435; 18:876902Þ;

m ¼ ðc 1Þ=ðcþ 1Þ;

U¼ ðv;u;SÞ;

V¼ ðv;0; SÞ:

ð4:1Þ

As indicated byTheorem 3.1, the equilibrium point Vof(2.14)is unstable and the point Vþis asymptotically stable The point V0is chosen to be very closed to the point V:

V0¼ Vþ ð0:001; 0; 0:0001Þ:

We will compute the trajectory of(2.14)with the parameters given by(4.1)starting at V0using the function ODE45 of MAT-LAB from y ¼ 0 to y ¼ 50.Figs 1 and 2show that this trajectory converges to the asymptotically stable equilibrium point Vþ

as y becomes larger and larger Furthermore,Fig 1reveals that the trajectory may enter the domain of attraction of the asymptotically stable equilibrium point Vþ

This test indicates that the traveling wave may exist If it exists, it is approximated by a reasonable trajectory which starts near the left-hand state Uand converges to the right-hand state Uþ

4.2 Test 2: Positive shock speed and high thermal conductivity

Let us consider numerical approximations of a traveling wave of(1.1)for an ideal fluid, where the shock is peed is positive and the thermal conductivity is low This traveling wave can be used to approximate a given shock wave between a left-hand state denoted by Uand a right-hand state denoted Uþ The parameters for this test are chosen by

c¼ 1:4;

S¼ 1;

c ¼ 1;

k¼ 1=10;

l¼ 5k;

j¼ 100;

ðv;u;S;pÞ ¼ ð1; 1; 1:5; 0:65948851Þ;

ðvþ;uþ;Sþ;pþÞ ¼ ð2; 0:3521771; 1:4588051; 0:239814Þ;

m ¼ ðc 1Þ=ðcþ 1Þ;

U¼ ðv;u;SÞ;

V¼ ðv;0; SÞ:

ð4:2Þ

In this case, the equilibrium point V of(2.14)is unstable and the point V is asymptotically stable

Fig 1 Test 1: approximate connection between the unstable equilibrium point V  and the asymptotically stable equilibrium point V þ A trajectory starting near the unstable equilibrium point V  approaches the asymptotically stable equilibrium point V þ