On Stability of Curvilinear Shock Wave in a Viscous Gas

International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186
DOI: 10.5923/j.ijtmp.20120206.02

On Stability of Curvilinear Shock Wave in a Viscous Gas

Alexander Blokhin1, Boris Semisalov2,*

1Sobolev Institute of M athematics SB RAS, 4 Acad. Koptyug avenue, 630090, Novosibirsk Russia
2Design Technological Institute of Digital Techniques SB RA S, 6 Akad. Rzhanov street, 630090, Novosibirsk Russia

Abstract The planar shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small

perturbations. As in the case of a planar shock wave we suggest such boundary conditions that the linear initial-boundary
value problem on the stability of a curv ilinear shock wave (subject to these boundary conditions) is well-posed. We also
propose a new effective computational algorith m for investigation the stability. This algorithm uses the nonstationary
regularizat ion, the method of lines, the stabilizat ion method, the spline function technique and the sweep method. Applying it
we succeed to obtain the stationary solution of the considered boundary-value problem justifying the stability of shock wave.

Keywords Stability of Shock Wave, Co mpressible Heat-Conducting Polytropic Viscous Gas, Navier-Stokes Equations,

Rankine-Hugoniot Conditions, Linearizat ion of Function, Nu merical Solutions, Stabilizat ion Method, Regularizat ion

1. Introduction limit ing one under vanishing dissipation[22–25]. Although,
it should be noted that until now such a viscous profile
The motion of continuous media is often acco mpanied by (continuous) approach was applying for different concrete
the formation of transitional zones of strong gradients, where models of continuum mechanics mainly in the
flow parameters (velocity, density, pressure, temperature, one-dimensional case. So, it cannot be fully considered as an
etc.) vary rapidly. If d issipative mechanis ms are neglected, alternative one to the discontinuous approach for
then such thin zones are usually treated as surfaces of strong mu lti-dimensional shock fronts. At the same time, for
discontinuity. In that case the flow parameters change continuous media with dissipation only the continuous
step-wise with ju mps on a propagating surface of strong approach has yet a sufficient justification (at least, on the
discontinuity (e.g., shock wave). Note that motions of ideal one-dimensional level[26]).
continuous media are usually described by hyperbolic
conservation laws for which the mathematical theory of The indirect confirmation of the correctness of precisely
shock waves has been well d iscovered not only for this continuous approach for v iscous conservation laws is the
one-dimensional[1–10] but also for mult i-d imensional following. Hyperbolic conservation laws modeling motions
flows[11– 21]. of ideal media have such a property that their solutions are
continuous and single-valued during short time only (even
Concerning continuous media with dissipation (e.g., for rather smooth initial data). After that the so-called
viscosity, heat conduction, etc.), transitional zones of strong gradient catastrophe occurs (see, e.g.,[10]), and one has to
gradients can also be formed, and there emerges the introduce strong discontinuities. Solutions of viscous
necessity for the mathematical modeling of such a conservation laws have apparently no such a property. This
phenomenon. In this paper, we are concerned with the is indirectly confirmed by nu merous results (e.g.,[27–30])
motion o f a viscous compressible gas in the framework o f the concerning global existence theorems for the Navier-Stokes
Navier-Stokes model. As is known, the Navier-Stokes eq u atio ns .
equations are applied for solving the problem on shock
structure in a v iscous and heat conducting gas (see, e.g., the In this connection, we especially refer to interesting
classical approach in[22]). In this problem, instead of a results in[30] where the global existence and uniqueness of
surface of strong discontinuity one considers a thin the weak solution of the one-dimensional Navier-Stokes
transitional zone (viscous profile) where flo w parameters equations written in the Lagrangian coordinates has been
vary continuously. proved for the case of discontinuous initial data. Under
certain natural restrictions on the initial data it was shown
A strong discontinuity in an ideal med iu m is said to have a that shock discontinuities do not arise in solutions of the
viscous profile (o r structure) if the discontinuous flow is a Navier-Stokes equations.

* Corresponding author: At the same time, it should be noted that in some works
[email protected] (Boris Semisalov) (their number is not small) the d iscontinuous approach is
Published online at http://journal.sapub.org/ijtmp nevertheless used for shock waves in a viscous gas. For
Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved example, to estimate the influence of a small viscosity to the
evolution of perturbations of planar gas dynamic shock

171 International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186

waves it is assumed in[31] that one can neglect the width of with a mathematical ground for steady-state calculations for
the shock layer. Therefore, as for an inviscid gas, the blunt body flows with dissipation. In[39,40], on the examp le
problem on the evolution of perturbations is reduced in[31] of the linearized stability problem for the shock discontinuity
to the study of a linear in itial boundary value problem with in a viscous gas the idea of such a mod ification was proposed
boundary conditions on a shock front obtained by the for the one-dimensional case. The essence of this idea is that
linearization of the generalized Rankine-Hugoniot relat ions. for the orig inal shock front problem one writes additional
Another examp le is the nu merical analysis of boundary conditions so that for the modified problem the
two-dimensional steady viscous flows near b lunt bodies[32] steady flow regime with a shock wave described above
(we just refer to[32] as to a typical paper fro m nu merous becomes asymptotically stable (by Lyapunov). That is, at
computational works relating to the subject under least on the linearized level this might justify the
discussion). To bound essentially the calculated do main, stabilization method wh ich can now be applied for finding
where solutions of the co mpressible Navier-Sto kes equations (e.g., nu merically) steady flow regimes for a viscous gas
are sought, one introduces a bow shock that is treated in[32] with a shock wave. The mentioned additional boundary
as a strong discontinuity on which surface the generalized conditions were suggested to be written with regard to an a
Rankine-Hugoniot conditions hold. Moreover, steady flows priori informat ion about steady-state solutions of the
were being computed in[32] by the stabilization method, i.e., Navier-Stokes equations being sought by the stabilization
steady-state solutions of the Navier-Stokes equations were method.

being sought as a limit of unsteady ones under t → ∞ . In the present paper we consider a curvilinear shock wave.

In[33–35], on the examp le of the co mpressible 2. Preliminary Information
Navier-Stokes equations the groundlessness of the approach
based on the consideration of shock waves in a v iscous gas as 2.1. The System Describi ng Gas Dynamics
fictit ious surfaces of strong discontinuity was shown. It
appears that this conclusion can be already drawn according Recall that in[33,39] the stability of shock waves in a
to the linear analysis. For this purpose one studies the initial
boundary value problem (IBVP) obtained by the compressible viscous gas described by the well-known
linearization of the Navier-Stokes equations and the jump
conditions with respect to their piecewise constant solution. Navier-Stokes equations was studied. Let us consider the
This piecewise constant solution describes the following
flow reg ime for a v iscous gas: a supersonic steady viscous same flo w regime as in[33,39] (see Fig. 1): the supersonic

flow (for x > 0 ) is separated fro m a subsonic one (for steady viscous flow is separated from the disturbed flow by a
x < 0 ) by a planar shock discontinuity (with the equation
x = 0 ). It was shown in[33–35] that this planar shock wave shock wave with the equation

is unstable depending not on the character of linearized x = f (t, y) (1)

boundary conditions at x = 0 . This is a direct consequence where t is the time, ( x, y) is the Cartesian coordinate

of the fact that the number of independent parameters system (in this paper we restrict ourselves to the
determining an arbitrary perturbation of the shock front is consideration of the planar case).
greater than that of the linearized boundary conditions. That
is, the shock wave in a viscous gas viewed as a surface of Figure 1. Viscous gas flow with a shock wave
strong discontinuity is like nonevolutionary (undercompress
ive) discontinuities in ideal med ia (see[36,37]). We write down the Navier-Stokes equations for a
compressible heat-conducting polytropic gas in the Cartesian
Mathematically, the exponentially increasing particu lar coordinate system in dimensionless form as (see[33,39,41])
solutions constructed in[33–35] for establishing linear
instability are, actually, Hadamard-type examples (see,
e.g.,[14,38]) that indicate the ill-posedness of the linearized
stability IBVP mentioned above. The discovered instability
can be also treated as an indirect proof of the inad missibility
of steady-state calculations for v iscous blunt body flows with
a bow shock discontinuity. Fro m the physical point of v iew,
this means the practical unrealizab ility of the steady flow
regime for a viscous gas described above.

At the same time, accounting for some advantages of the
d is co n t in u o us ap p ro ach (es p ecial ly fo r n u me r ical
calculations), one would like to modify this approach so that
it might be applied (together with the stabilization method)

Alexander Blokhin et al.: On Stability of Curvilinear Shock Wave in a Viscous Gas 172

ρt + div(ρu) = 0, u = (u, v),  where M∞ = u∞ is the Mach number for the
1
1, 
{ }(ρu)t + div(ρuu) + px = Re∞  (γ p∞V∞ )2
(ησ xx )x + (ησ xy ) y
 (2) upstream flo w, ω (0,5 ≤ ω ≤ 1) is a constant.

{ }(ρv)t + div(ρvu) + py = 1, 
(ησ xy )x + (ησ yy ) y Re∞  2.2. Rankine-Hugoniot Conditions

(ρ E )t + div [(ρ E + p)u] = γ div(η∇T ) + η Φ. On the surface of shock wave (1) generalized
Re∞Pr Re∞
Rankine-Hugoniot conditions hold (see[33,39]). Using some

1 cumbersome man ipulations they can be reduced to the
ρ
Here ρ is the gas d ens ity , V = is the specific following

ft = u − fyv −V , (4)
1−V
volume, u, v are the Cartesian co mponents of the velocity
vector u , p is the pressure,

E = T + u2 + v2 , ( )( )(1− u +
2 divu = ux + vy , f yv)2 + (V − 1) 1+ f 2  − lˆ = 0, (5)
y

σ xx = 2ux + 1divu = 3ux + 1vy ,σ xy = (uy + vx ) , 2 ( pV )ω
Re∞
σ yy = 3vy + 1ux , v + f yu − fy = , (6)

1 =  ζ − 2  , 3 =  ζ + 4  lˆˆ + (V  +l
 η 3   η 3  −1 2
T − −1) =
are constants, γ
(7)
η , ζ are the first and second viscosity coefficients )2
=1 (v + fyu − fy γ 2 ( pV )ω
(s ee[39 ,4 1] ), 2 + Re∞Pr( ft −1) ( f yTy − Tx ).
1+ f 2
u∞ Lˆ ρ ∞ y
η∞
Re∞ = is the Reynolds number (see[41]), Here

pV ηCP  = p − 2 ( pV )ω 1divu + 2 (ux + vy f 2 − f y (uy + vx ))  ,
γ −1  Re∞ y 
T = is the temperature, Pr = is the 1+ f 2 
y

Prandtl nu mber (see[41]),  = (uy + vx )( f 2 − 1) + 2 f y (vy − ux ) .
y
 is the heat conductivity,
ft −1
= P ;
γ V P, V are the specific heat capacities,

Φ = 2(ux )2 + 2(vy )2 + 1(divu)2 + (σ xy )2 ; 2.3. The Problem Written in the Curvilinear Coordi nate
S ys tem

the following scaling factors were used to define the Further we will need the Navier-Stokes equations (2)

dimensionless time t , the coordinates x , y , the density rewritten in orthogonal curvilinear coordinates (α , β ) :

ρ , the velocity co mponents u , v , the pressure p (see α = x − y2 , β = yex. (8)
2
Fig. 1): Lˆ Lˆ , ρ∞ , u∞ , ρ∞u∞2 , where Lˆ is a
α and β in (8) are the d imensionless coordinates.
,

u∞

characteristic length (see also Remark 2.1); the Formulae (8) can be rewritten in the dimensional form as

dimensionless viscosity coefficients η , ζ are defined by α = x − y2 , β = yex/Lˆ , (8')
2Lˆ
using η∞ as the scaling factor; the viscosity coefficient η
where Lˆ > 0 is a characteristic length (see Remark 2.1).
depends on pV according to the power law (see[41])

η  pV ω )ω ,  The curvilinear coordinates α , β are chosen so that the
=  p∞V∞ =
=   2 ( pV  front of the shock wave is described in the stationary case by

 (3) the equation (see Fig. 2)
,
2 (γ M 2 )ω =  1 ω , lˆ = 1 α = 0, i.e. x = f ( y) = y2 . (1')
∞  lˆ  2
γ M 2
∞

173 International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186

where

N = lnV = − ln ρ ;

Vα , Vβ are the physical components of the velocity

vector u in the curvilinear coordinates (α , β ) related to
the Cartesian components u , v as

Vα = Hαvα = Hα (u − yv), (13)
Vβ = Hαvβ = Hα ( yu + v),

where vα , exvβ are the so-called contravariant
components of the velocity vector u (see[42]);

divu = R + Q, (14)

Figure 2. Front of the shock wave (Vα )α (Vβ )β
Hα Hβ
Remark 2.1. Note that the radius of curvature of the R = − Vβ ⋅ yH 3 , Q= − Vα Hα3 ,
α
stationary shock wave at x = y = 0 can be taken as the

characteristic length Lˆ Λ = (Vα )β + (Vβ )α + Hα3 (Vβ + yVα ),
Hβ Hα
We rewrite the Navier-Stokes equations in the coordinates

(α , β ) in the fo llo wing nonconservative form (see[41]):

Nt + Vα (N )α + e xVβ (N )β − divu = 0, (9) Φ = 2R2 + 2Q2 + Λ2 + 1(R + Q)2, (15)
Hα Hα
1 , Hβ = e−xHα are the La mé coefficients
Hα = 1+ y2

(Vα )t + Vα (Vα )α + e xVβ (Vα )β +V ( p)α + (s ee[42 ]),
Hα Hα Hα
(...)α = ∂ (...) , (...)β = ∂ (...) .
∂α ∂β
Hα3Vβ (Vβ − yVα ) = V 2  (( pV )ω (3R + 1Q))α (10)
Re∞  Hα
 Remark 2.2. Taking into account formu lae (13), we can
rewrite equations (9)–(12) as
+ (( pV )ω Λ)β + 2Hα3 ( pV )ω (Q − R − yΛ )  ,
Hβ  Nt + vα (N )α + exvβ (N )β − divu = 0, (9')

e xVβ (vα )t + vα (vα )α + exvβ (vα )β + V ( p)α +
Hα Hα2
(Vβ )t + Vα (Vβ )α + (Vβ )β + Vex ( p)β −
Hα Hα = V2  (( pV )ω (3R + 1Q ))α + (10')
+Hα4 (vβ − yvα )2 Re∞ 
pV )ω  H 2
−Hα3Vα (Vβ − yVα ) = V2  (( Hα Λ )α + (11) α
Re∞ 
 + (( pV )ω Λ)β ex + 2( pV )ω H 2 ((Q − R − yΛ)  ,
Hα2 α 
+ (( pV )ω (3Q + 1R))β  
Hβ − 2 H 3 ( pV )ω ( y(Q − R) + Λ ) 
α


e xVβ (vβ )t + vα (vβ )α + exvβ (vβ )β + Ve x ( p)β −
Hα Hα2
Tt + Vα (T )α + (T )β + pVdivu = γV2 
Hα Re∞Pr V 2  (( pV )ω Λ )α
−Hα4 (vβ − yvα )(vα + yvβ ) = Re∞  Hα2 +

  1 (( pV )ω (T )α 1 (( pV )ω (T )β (12) (11')
 Hα Hα Hβ Hβ
 )α + )β − ex
Hα2
+ (( pV )ω (3Q + 1R))β −

( pV )ω (T )α Hα2 + ( pV )ω (T )β β H 2  + V 2 ( pV )ω Φ, Hα2 )ω 
α  Re∞ 
2 ( pV ( y (Q − R) + Λ ) 

Alexander Blokhin et al.: On Stability of Curvilinear Shock Wave in a Viscous Gas 174

Tt + vα (T )α + exvβ (T )β + pVdivu = γV2  constant behind the shock wave α = 0 :
Re∞Pr
η = η* = 2 ( p*V *)ω , (3')

 1 (( pV )ω (T )α )α + 1 (( pV )ω (T )β )β − (12') where p* , V * are constants;
 Hα Hα Hβ Hβ
b)  = 0 , i.e., the Prandtl number Pr = ∞ ;

}( pV )ω (T )α Hα2 + ( pV )ω (T )β β Hα2 + V 2 ( pV )ω Φ, c) vβ = ywβ , where wβ is a new dependent variable;
Re∞
d) the functions vα , wβ are denoted as v, w.
Here
Then in the region 2+ = {(α, β ) : α > 0,| β |< ∞}
R = (vα )α − yH 4 (vβ − yvα ) ,
α
the system(9')–(12') in the stationary case can be rewritten as
Q = ex (vβ )β − Hα4 (vα + yvβ ) ,
v(V )α + β w(V )β −V {(v)α +
Λ = ex (vα )β + (vβ )α + Hα2vβ ; (16)
}+β (w)β + Hα4 (1 − y2 )(w − v) =0,
the values of divu , Φ are calculated by using (14),
{v( p)α + β w( p)β + γ p (v)α + β (w)β + (17)
(15).
Remark 2.3. We are also interested in stationary solutions }+
H 4 (1 − y2 )(w − v) = χ (γ −1)Φ,
to equations (9)–(12) (or (9′) – (12′) ) with boundary α

conditions set at the front of the shock wave α = 0 (see v(v)α + β w(v)β + V ( p)α + y2 Hα4 (w − v)2 =
Hα2
Fig. 2). In the stationary case conditions (4)–(7) take the
following form:

V = vα , (4') = V χ  3 ((v)α − y2 Hα4 (w − v))α +
 Hα2
1  (18)

(1 − vα ) = H 2 ( − lˆ), (5') + Hα4 ex (ex (v)β
α Hα2
(w − v))α + + y(w)α )β +

(vβ − y) = 2 (PV )ω Λ, (6') +2Hα2 (β (w)β + Hα2 (w − v) (v)α − β (v)β − y2 (w)α  ,
Re∞ 


lˆˆ −1)  + l v(w)α + β w(w)β + Hα2w(w − v) − Hα4 (w − v)
−1 2
T − + (vα = (v y2w) Ve2 x  1 
γ β Hα2  yH
(7') + + ( p)β =Vχ  2
α
Hα2 (vβ − y)2 + γ 2 ( pV )ω 1
= 2 Re∞Pr Hα2 (T )α , (ex (v)β ex3 Hα4 (w
yHα2
+ y(w)α )α + (β (w)β + − v))β +(19)

where = p − 2 ( pV )ω (3R + 1Q ) . + e x 1 ((v)α − y2Hα4 (w − v))β − 2Hα2 (β (w)β +
Re∞
yH 2
α

+ H 2 ( w − v) − (v)α + (w)α + e2x (v)β ) 
α β 

3. Solving the Stationary Navier-Stokes
where χ = η* ,
Equations in a Neighborhood of the Re∞

Line β = 0 ( y = 0) Φ = 2((v)α − y2Hα4 (w − v))2 + 2(β (w)β +
+Hα4 (w − v))2 + (ex (v)β + y(w)α )2 +
3.1. Additi onal Assumpti ons
The system of equations (9')–(12') is too comp licated. Let +1((v)α + Hα4 (1− y2 )(w − v) + β (w)β )2.

us modify it in a certain way in the stationary case. To For α = 0 the solution of system (16)–(19) should
simp lify further calculat ions we will assume that
satisfy relations (4')–(7') which under assumptions a) – d)
a) the v iscosity coefficient η (see formula (3)) is cited above can be rewritten as follows:

V = v, (20)

175 International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186

(1 − v) = 1 ( − lˆ), (21) v0w '0 + (w0 − v0 )2 + 2V0e2α P1 = V0χ{−w '0 +
Hα2 +w ''0 + 2e2α v1 + 2e2α v1′ + 3e2α (4w1 + 2(w1 − v1) − (29)
−4e−2α (w0 − v0 )) + 1e2α (2v1′ − 2e−2α (w0 − v0 )) −
(w − 1) = χ ((w)α + e2x (v)β ), (22)
β −2(w0 − v0 − v0′ + w '0 + 2e2αv1)},

− 1)(v − 1)  + lˆ (w − 1)2 for 0 < α < ∞ ;
2 2
pV − lˆ + (γ = Hα2 y2 . (23) V0 = v0, (30)
1 − v0 = 0 − lˆ, (31)

Here w0 − 1 = χ (w '0 + 2v1), (32)
(33)
{ = p − χ 3((v)α − y2Hα4 (w − v))2 + P0V0 − lˆ + (γ − 1)(v0 − 1) 0 + lˆ = 0
2
}+ 1(β (w)β + Hα4 (w − v)) , x = x(0, β ), y = y(0, β ).

In addition, we will assume that the functions v , w for α = 0 ;
satisfy the so-called ''soft condition'' (see[41]) as α → +∞ :
v0′ , w '0 → 0 where α → ∞. (34)
(v)α , (w)α → 0. (24)
Here

3.2. The Tayl or Expansion with Res pect to the Argument Φ0 = 2(v0′ )2 + 2(w '0 − v0′ )2 + 1(v0′ + w0 − v0 )2,
β 0 = P0 − χ{3v0′ + 1(w0 − v0 )},

Further we will simp lify the nonstationary system v0′ (α ) = d v0(α ),
dα
(9')–(12') using stationary solutions v , w in a
neighborhood of the line β = 0 for definit ion of system’s ect.
coefficients. The functions V , p , v , w are even with Remark 3.2. As fo llows fro m (26)–(34), for the
respect to the argument β (see Fig. 2). Therefore, in a
determination of the functions V0 , P0 , v0 , w0 we have
neighborhood of the line β = 0 we will search them in the
to know the functions v1 , P1 , w1 . Further, assuming in
form of the following series (see[43,44]):
these relations that v1 ≡ 0 , P1 ≡ 0 , w1 ≡ 0 , we will

consider an approximate solution of the problem (26)–(34).

V (α, β ) = V0(α ) + V1(α )β 2 + ...,  Remark 3.3. We set in formu la (3 ' ) that p* = P0 (0) ,
p(α , β ) = P0(α ) + P1(α )β 2 + ...,
v(α , β ) = v0(α ) + v1(α )β 2 + ...,  (25) V * = V0 (0) . Fro m (33), (31) we have:

 P0 (0)V0 (0) = lˆ + (γ − 1)∆ 2lˆ + ∆, ∆ = 1 − v0(0). (35)
2
w(α, β ) = w0(α ) + w1(α )β 2 + .... Let also in formulae (3), (3 ' ) ω = 1. Then, it follows

Remark 3.1. Using formu lae (8), we can also expand the fro m (30), (31), (35) that

functions x(α, β ) , y(α, β ) into series with respect to P0 (0) = 1 lˆˆ+ γ −1 ∆(2l + ∆)  , 
v0 (0)  2  
the variable β . 
(lˆ + ∆)(1− ∆) 
Substituting expansion (25) into (16)–(24) and equating 1 − χˆ (1 − ∆)[3v0′ (0) + 1(w0 (0) − v0 (0))] =  (36)

the coefficients at the same powers of beta β , we obtain a

relation for determining the functions V0 , P0 , v0 , w0 , = lˆˆ+ γ −1 ∆(2l + ∆), 

etc. in series (25). Equating the coefficients at β 0 , we 2

have: χˆ = 2 1
Re∞ lˆRe∞
v0V0′ −V0{v0′ + w0 − v0} = 0, (26) where = .

v0P0′ + γ P0{v0′ + w0 − v0} = χ (γ − 1)Φ0, (27) 3.3. The Boundary-Value Problem of Second Order for
the Components of Velocity Vector
v0v0′ + V0P0′ = V0χ{3v0′′+ 1(w '0 − v0′ ) + (28)
+2e2αv1 + w '0 + 2(w '0 − v0 − v0′ )}, Let us return to the boundary-value problem (26)–(34).
Fro m (26), (27) we have:

Alexander Blokhin et al.: On Stability of Curvilinear Shock Wave in a Viscous Gas 176

V0 (α ) = v0 (α )I (α ),  perturbations and freeze its coefficients in the neighborhood

P0 (α ) = P0 (0) Vv00((α0)) γ +  (37) of the line β = 0 at the point α = 0 . Finally we wil l
 
  obtain the simp lified nonstationary system and seek its

∫+ χ (γ − 1) 1 α  Φ0 V0γ  (s)ds.  2 :α > 0,| β |< ∞
V0γ (α ) 0  v0  s tatio nary solution in the region + .
 

Here Note that the variab les of the system obtained still depend

(α∫I ) = exp  α  w0 − v0  ( s )ds  , χ = P0(0)v0(0)χˆ , on β ∈ (−∞, +∞) . Further in section 6 using a new
 0  v0  
    effective co mputational algorith m we will find out that the

and the quantity P0 (0) can be expressed in terms of solution of considered problem for small perturbations

v0 (0) by formula (36). Let us rewrite equations (28), (29) vanishes as t → ∞ . This is an evidence of the stability of

as follows: shock wave.

v0′′(α ) = F (v) = v0′ (α ) − 3 −1 w '0(α ) − 2  4.1. The Linarization of Nonstationary Equati ons
3 3 (9')–(12')
(38)
Let us linearize of system (9')– (12') about the basic
(w0(α ) − v0(α )) + 1 P0′(α ) + 1 v0′ (α ), solution (25) and freeze then the coefficients of the
χ3 I (α )χ3
linearized system on the line β = 0 . Denoting small
w ''0(α ) = F (w) = 3w '0(α ) − 2v0′ (α ) + 2(33 − 1)
perturbations by the same letters, we finally obtain:
[w0 − v0 ](α ) + [w0 − v0 ]2 (α ) + w '0(α ) , (39)
V0(α )χ I (α )χ LV −V0(α )D −Vd0(α ) + vαV0′(α ) = 0, (43)

where d0 = v0′ + w0 − v0 , Lp + γ P0 (α )D + γ pd0 (α ) + vα P0′(α ) =

P0′(α ) = −γ  d0  (α ) + χ (γ − 1)  Φ0  (α ) , 0 < α < ∞. = (γ −1)χ 2 {4v0′ (α )ξ vα + 4[w0 − v0 ](α ) (44)
 P0 v0   v0 
    }(ζ vβ − vα ) + 21d0 (α )D ,

The boundary conditions for equations (38), (39) follow Lvα + V0(α )ξ p + vαa0(α ) −

fro m (36), (32), (34): −V= vV0v00′  (α ) V0(α )χ {3ξ D + ζΩ}, (45)

3v0′ (0) + (3 − 2)[w0 − v0 ](0) = Q =

∆ lˆγ + γ +1 ∆ −1 (40)
χˆv0 (0) 2
,
lˆˆ+ γ −1 ∆(∆ + 2l
2 ) Lvβ +V0(α )ζ p + vβ b0 (α ) =
= V0
(α )χ {3ζ D − ξΩ}. (46)

χ w '0(0) = w0(0) − 1, (41) Here

w '0(α ), v0′ (α ) → 0 as α → ∞. (42) L =τ + v0(α )ξ , τ = ∂, ξ = ∂ ,
∂t ∂α

For searching appro ximate solutions of the eα ∂
∂β
boundary-value problem (38)–(42) a new co mputational ζ = , D = ξ vα + ζ vβ − vα ,

algorith m was proposed. It is described in the section 5,

where we also present results of numerical experiments. Ω = ζ vα + vβ − ξ vβ , d0 = v0′ + w0 − v0 ,

4. The Linearization of Nonstationary a0 = v0′ + 2V0χ , b0 = w0 − v0 + 2V0χ3 .
Problem for System (9')–(12')
System (43)–(46) is considered in the domain
The nonstationary problem (9') – (12'), (4') – (7') is
rather complex, that is why further we will follow the ideas 2+ : α > 0,| β |< ∞ (at infinity the sought functions
fro m [14] and consider the simp lified particular case of it.
We are going to derive the linearized system for small tends to zero).
Let us simplify system (43)–(46) by freezing the functions

v0 (α ) , V0 (α ) , w0 (α ), P0 (α ), v0′ (α ), V0′(α ) an d

177 International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186

P0′(α ) at the point α = 0 and making the change of γ lˆ + γ −1 − γ + 1 v02 (0)
2 2
t = v0 (0)t (below we drop t ilde and again write t εˆ = γ − 1 γ 1 ,
lˆˆ+ˆ 2 −1 ∆(∆
instead of t ). We get finally that γ (1 − v02 (0)) l + 2 + 2l )

LV − D + d1(vα −V ) = 0, (43') dˆ = v02 (0) .
−1 (1 −
1 γ lˆ + γ 2 v02 (0))
M2
Lp + D + γ d1 p + d 2 vα = 2(γ −1)χ  Remark 4.1. According to [33,39] for the determination

{ } 2a2ξ vα + 2(d1 − a2 )(ζvβ − vα ) + 1d1D , (44') of the function F (t, β ) describing the small perturbation

Lvα + ξ p + a1vα − a2V = (45') of the shock front α = 0 we will add to (43')–(46'),

{ }= χ 3ξ D + ζΩ + 3ζvβ , (47)–(50) the equation in the form (see, for examp le,[33,39])

Ft = µ p |α=0,
where µ is a constant.

{ }Lvβ + ζ p + b1vβ = χ 3ζD − ξΩ − ζvα , (46') Remark 4.2. We also have to specify initial data for
system (43')–(46') with the last equality at t = 0 ,
(α , β ) ∈  2 = {(α, β ) : α > 0,| β |< ∞}
where + . We use data

L = τ + ξ , ζ = ∂ , D = ξ vα + ζvβ − vα , of the form
∂β
=vα (0,α , β ) f=0 (α ), vβ (0,α , β ) 0,
Ω = ζvα + vβ − ξ vβ ,  =− f0d(ˆα ) , V (0,α , β
 p(0,α , β ) ) =f0 (α ),

d1 = a2 +  w0 − v0  (0) , a2 =  v0′  (0) , where 0 ≤ f0 (α ) ≤ 0.001.
 v0   v0 
    Remark 4.3. We also assume that the small perturbations
vα , vβ → 0 as α → +∞ .
 v02   v0  (51)
γ P0V0   γ P0 
M2 =  (0) =   (0) , (see (24)).
In the section 5 a new computational algorith m for finding
d2 =  P0′  (0) = − 1 d1 + χ γ −1 1 Φ 0 (0) ,
 v0  M2 γ M2 stationary approximate solutions of the initial-boundary
  value problem (43 ' )–(46 '), (47)–(51) is described, where
we also present results of numerical experiments.

a1 = a2 + 2χ , b1 = d1 − a2 + 2χ3 ,

{ }Φ0 (0) = v02 (0) 2a22 + 2(d1 − a2 )2 + 1d12 . 5. A New Computational Algorithm for
Solving Boundary-Value Problems of
4.2. The Linearization of Stationary Boundary Second Order
Conditi ons (4')–(7')
Now we will propose a new numerical algorith m for

Let us now formu late boundary conditions at α = 0 . To solving the problem (38)–(42). In nu merical calcu lations we

this end we linearize conditions (4')–(7') posed at pass from infinite segment 0 < α < ∞ to a finite one and

α = F (t, β ) , where the function F is the small assume that 0 < α < L , where L is a large enough

perturbation of the shock front which unperturbed position is number. We make the change of variable x = α , 0 < x < 1 .
L
α = 0 . Freezing the coefficients at β = 0, we will finally
Let ξ = ∂ (do not forget that ξ= ∂ ) , then
obtain (α = 0) : ∂x ∂α

V = vα , (47) ξ = ∂ = 1 ξ, ∂2 = 1 ξ2 . Note a ls o that L has no
vα + dˆp = 0, (48) ∂α L ∂2α L2

χˆˆ3ξ vα + 1((vβ )β − vα ) = ε p, (49) physical meaning and dimension and the new notation x is

vβ = χ (vα )β + ξ vβ + vβ . (50) not equal to that from section 2 and concerns only sections 5

and 6.

Here 5.1. The Nonstati onary Regul arizati on and the
Stabilization Method

Alexander Blokhin et al.: On Stability of Curvilinear Shock Wave in a Viscous Gas 178

Introducing the notations w ( x) = w0 ( x) − 1, Remark 5.1. The main idea of the stabilization method is
v( x) = v0 ( x) − 1 , we rewrite the boundary-value
described, for example, in[47]. In our case for the
problem (38)–(42) as fo llo ws:
implementation of the stabilizat ion method we should

discretize the nonstationary equations (55), (56) by t ime and

ξ 2v  (v ) L[ξv 3 − 1ξw ] 2L2 ∆ 1   perform nu merical computations until the solution
3 3 3χ 
= = − − + ''stabilized''. In other words, the algorith m using this method

ξv  will stop only when the norm of the difference of the
I ( x)
L −γ P0( x)d0( x) χ (γ Φ0( x)   solutions at next and previous time layers beco mes small
 v(x) + 1 v ( x) +1  
L + − 1)  ,  (52) enough. Thus, we search a solution of problem (52)– (54) in

ξ2w =  (w ) = L[3ξw − 2ξv] + 2L2(33 − 1)∆ +  the form of a limit as t → ∞ of the solutions of the

nonstationary equations (55), (56) with the boundary

+L2 ∆ 2 + L ξw  conditions (53), (54).
χV0(x) χ I (x) 
For the application of the stabilizat ion method it is

necessary to specify init ial data for system (55), (56). Further

with the boundary conditions we assume that

ξv(0) + L 3 − 2 ∆ (0) = v(0) f (v (0)),  v(0, x) = γ 2 1 (lγ − 1),  (57)
3 + 
 (53)
 w (0, x) = 0, 0 < x < 1. 
ξw (0) = L w (0), 
χ  Using the idea o f the method of lines (see[45,46]), we

at x = 0 and ξv(1) = 0, discretize with respect to t equations (55), (56). To this end

ξw (1) = 0  (54) we introduce the notations

vn (x) = v(n∆, x) = ϕ, vn+1(x) = Φ , n = 1, 2... ,
w n (x) = w (n∆, x) =ψ , w n+1(x) = Ψ , n = 1, 2... ,

at x = 1. Here where ∆ is the time-step of the greed.
∆ ( s) Approximating the derivatives τ v , τ w in (55), (56) by
∫I ( x) = exp  L x (s) + ds  , Φ − ϕ Ψ −ψ
 0 v 1 
 
the expressions and respectively, we
∆∆
 v(0) + 1 γ
P0( x) = P0 (0)  V0( x)  + obtain ξ2Φ = Φ +  (ϕ) ,
 
(58)

1 x vΦ+01]( ξ2Ψ = Ψ +  (ψ ) , (59)
(x)L
∫+ χ (γ [ s)ds,
− 1) V0γ
0 where

Φ0(x) = 2(ξv)2 + 2(ξ∆ )2 + (3 − 2)(d0(x))2,  = 1 ,  (ϕ) = ξ2ϕ − ϕ + ∆ (v) ,
1+ ∆ 1+ ∆
V0(x) = [v(x) + 1]I (x)
L(1 − lγ + (γ + 1)v(0) / 2)  (ψ ) = ξ2ψ −ψ + ∆ (w ) .
f (v(0)) = 3 χ(1 + v(0))[l − (γ −1)v(0)(2l − v(0)) / , 1+ ∆
2]

d0(x) = ξv + L∆, ∆ = w − v. 5.2. The S pline-Function Techni que

For searching appro ximate solutions of the We will seek a solution of equations (58), (59) in the form

boundary-value problem (52)–(54) we will use the of cubic  2 interpolation splines (see[45,46,48]). For

stabilization method. To this end let us apply to the equations example, let us write the appro ximate solution of equation

of system (52) the nonstationary regularization proposed in (58) in the fo llo wing way:

the[45,46]. As a result, we obtain the systemof nonstationary S (x) = (1− κ )Φ k + κΦ k+1 −

equations (1 − ξ2 )τ v = ξ2v −  (v), h2 (60)
6
(55) − κ (1− κ )[(2 − κ )mk + (1 + κ )mk +1 ],

(1 − ξ2 )τ w = ξ2w −  (w ). (56) where

Here τ = ∂ and t plays the role of time, i.e., we κ = x − xk ,∈[xk , xk+1], xk = kh, k = 0, K − 1,
∂t h
Kh = 1, Φ k = Φ ( xk ), mk = ξ2Φ ( xk ).
assume further that the unknowns v , w depend also on
the varia=ble t : v v=(t, x), w w (t, x) . The cubic spline (60) should be continuous together with

179 International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186

its first derivative on the whole segment [0,1] (the second {1 − h2 }Ψ k−1 − 2{1 + h2 }Ψ k +
6 3
derivative is continuous according to the definition of spline (63)
(60)). h2 h2
+{1 − 6 }Ψ k+1 = 6 {k(−ψ1 ) + 4k(ψ ) +  (ψ ) },
The first derivative of the cubic spline has the following k +1
form:
k = 1, K − 1 for finding a solution of equation (59),
ξS( x) = Φ k+1 − Φ k −
h where  (ψ ) =  (ψ ) (xk ). The derivatives
k
h ξw n+1(x) = ξΨ (x) can be found as follows:
− 6 [(2 − 6κ + 3κ 2 )mk + (1 − 3κ 2 )mk+1].

Then, computing the aggregates ξΨ ( xk+1) = Ψ k+1 − Ψ k + h [mk + 2mk+1],
h 6
ξS(xk + 0), ξS(xk − 0),
x ∈[xk , xk+1], k = 0, K − 1,
where Ψ 1 − Ψ 0
Φ k+1 − Φ k h
ξS ( xk + 0) = h − h [2mk + mk+1], ξΨ (0) = − h [2m0 + m1 ].
6 6
Φ k − Φ k−1 Replacing in (53), (54) the derivatives ξv(0), ξw (0) ,
ξS( xk − 0) = h + h [mk −1 + 2mk ], ξv(1) , ξw (1) by their difference analogues
6

and equating them, one gets Φ 1 − Φ 0 , Ψ 1 − Ψ 0 , Φ K − Φ K−1 , Ψ K − Ψ K−1 ,

1 mk −1 + 2mk + 1 mk +1 = 3 (Φ k+1 − 2Φ k + Φ k−1). (61) hh h h
2 2 h2
we obtain the boundary conditions on the (n + 1) th time
Here k = 1, K −1.
layer
Assuming in (58) x = xk and substituting mk fro m
[1 + hL 3 − 2 + hf (ϕ0 )]Φ 0 = Φ 1 + hL 3 − 2 ψ , 
(58) into (61), one gets the follo wing three-po inted 3 3 
0  (64)

difference scheme: Φ K = Φ K−1 

{1 − h2 }Φ k−1 − 2{1 + h2 }Φ k + for scheme (62) as well as the boundary conditions the on
6 3
(62) (n + 1) th layer
h2 h2
+{1 − 6 }Φ k+1 = 6 {k(−ϕ1) + 4k(ϕ ) +  (ϕ ) }, Ψ 0 1 + hL −1 Ψ 1, 
k +1  χn  
=  
k = 1, K − 1,  (ϕ ) = k(ϕ ) (xk ). (65)
where k
Ψ K = Ψ K−1 
The derivatives ξvn+1(x) = ξΦ (x) at the grid nodes 

(i.e., at the points x = hk = xk , k = 0, K ) are required for for scheme (63). Here −)[llγ−+(γ(γ−+1)1ϕ)ϕ0(02/l2−)ϕ0

computing the right-hand sides  (v) and  (w ) on the f (ϕ0 ) = 3 χ (1 L(1 ) / 2] ,
+ ϕ0
next t ime layer. These derivatives can be computed using

spline (60). ϕ0 = ϕ(0),ψ0 =ψ (0).

Since Φ k+1 − Φ k
h
ξS(x) = − h [(2 − 6κ + 3κ 2 )mk + (1 − 3κ 2 )mk+1], 5.3. The Use of S weep Method
6
Equations (62), (63) with the boundary conditions (64),
where x ∈[xk , xk+1], k = 0, K − 1 , then
Φ k+1 − Φ k (65) can be solved by using the sweep method. Let us
h
ξΦ ( xk+1) = h describe the application of this method on the example of
6
+ [mk + 2mk+1], problem (62), (64). The co mputations for scheme (63) with

the boundary conditions (65) can be done in the same

x ∈[xk , xk+1], k = 0, K − 1, manner.

ξΦ (0) = Φ 1 − Φ 0 − h [2m0 + m1 ]. We write down system (62) in the form
h 6
Φ k−1 − Φ k + Φ k+1 = k , k = 1,..., K −1. (66)

Using similar argu ments, we get the three-pointed The boundary conditions (64) are rewritten as follo ws:
Φ 0 = A1Φ 1 + B1,
difference scheme (67)

Alexander Blokhin et al.: On Stability of Curvilinear Shock Wave in a Viscous Gas 180

Φ K − G1Φ K −1 = G2 , (68) account the obtained vn+1( xk ) and solve the

where boundary-value problem (56), (53), (54). At last, we get the

 =  = 1 − h2 , = 2(1 + h2 ), solution vn+1(xk ), w n+1(xk ), k = 0,..., K of system (55),
63
(56) at the (n + 1) th time layer. Then, we pass to the next
k = h2 {k(−ϕ1) + 4k(ϕ ) +  (ϕ ) },
6 k +1 time layer.
According to the idea of the stabilization method (see
A1 = 1 + hL 3 − 2 + hf (ϕ0 −1 ,
 3 ) Remark 5.1), these actions should be repeated until the
 solution becomes ''stabilized''. The obtained values of
unknowns are exact ly the desired approximate solution of
B1 = 1 + hL 3 − 2 + hf (ϕ0 −1 hL 3 − 2 ψ 0 = 1, the boundary-value problem (52)–(54).
 3 ) 3
 For the organization of co mputations by the proposed
scheme it is necessary to specify the list of physical and
G1 = 1, G2 = 0. numerical parameters. In Table 1 we give the description of
each parameter and the range of its values.
Remark 5.2. Problem (63), (65) can be also easily

reduced to the problem of form (66)–(68).

According to the idea of the sweep method we search a

solution of (66)–(68) as

Φ k = Ak+1Φ k+1 + Bk+1, k = 0,..., K −1. (69)

Substituting (69) into (66), we find recurrent fo rmulae for

the sweep coefficients Ak , Bk , k = 1,..., K :

Ak +1 = ( − Ak )−1, Ak  (70)
Bk +1 = (Bk  + k )( − )−1.

For k = 1 we take the values of the sweep coefficients

A1, B1 fro m the boundary condition (67). Thus, in the cycle

of the forward sweep we can find the coefficients

Ak , Bk , k = 1,..., K by formu las (70). Then, using the

boundary condition (68) and formu la (69), we can co mpute

the solution at the point k = K :

Φ K = G1BK + G2 .
1− G1AK
At last, knowing Φ K and the coefficients of the sweep

method, we can find a solution of problem (62), (64) by

formula (69) in the cycle of the backward sweep.

Remark 5.3. Using the same arguments, we can find a

solution of problem (63), (65) by applying again the sweep

method (see Remark 5.2).

5.4. The Computati onal Scheme and Numerical Results

As a result, the scheme of searching appro ximate solutions
of system (55), (56) at each time layer is the following.

Starting fro m the values of unknowns obtained on the n th

time layer we co mpute the values of variables of the problem

and the right-hand side  (v) ( xk ) at each point of the
space greed. Then we find the solution vn+1( xk ) of

equation (55) with boundary conditions (53), (54) by using
the sweep method. After that we co mpute the values of

variables and right-hand side  (w ) ( xk ) by taking into

181 International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186

6. Numerical Results Showing the
Stability of Shock Wave

6.1. The Fourier Transform for Nonstati onary Problem
(43')–(46'), (47)–(51)

Now let us describe the algorithm of searching

approximate stationary solutions of equations (43')–(46') for

small perturbations with the boundary condition (47)–(51).

As it was made in the section 5 the stationary solution will be

find with a help of the stabilizat ion method. Following

section 5, we assume that 0 < α < L , where L is larg e

Figure 3. The functions v( x) and w ( x) and their derivatives enough, replace the variable α by x = α , (0 < x < 1)
obtained after computations with the parameters γ = 1.4 , Re = 150 , L
M = 4 , 3 = 2 , L = 10 , K = 1000, ε1 = 10−4 , ∆ = 0.001
and denote ξ = ∂ , τ = ∂ (do not forget that ξ = ∂ ).
Table 1. Physical and NumericalParameters ∂x ∂t ∂α

Then

Quant ity Descript io n Value ξ = ∂ = 1 ξ, ∂2 = 1 ξ2 .
polytropic exponent ∂α L ∂2α L2
γ 1.4
Assume that at t = 0 the unknowns of problem (43

Re Reynolds number 0 ÷ 1000 ' )–(46 ') satisfy initial data (see Remark 4.2).
Remark 6.1. In the section 4 the computational algorith m

for searching the values of functions v( x) and w ( x) was

M Mach number 2, 4 described in detail. In th is case the functions v0 ( x) and

3 coefficient 2, 3 w0 ( x) needed for the determination of the coefficients of

equations (43')–(46') can be expressed as follows:

v0(x) = v(x) + 1, w0(x) = w (x) + 1.

L 0<α < L 1 ÷ 30 Let vˆβ (t, x,ω) be the Fourier transform of the velocity

K number of nodes of the 100 ÷ 3000 component vβ with respect to the variable β , where ω
space greed
is the Fourier parameter (the notation ω corresponds only
to this section, in other sections ω is a constant from (3))::

∆ step of the time greed 0, 01 ÷ 10−6 ∫vˆβ (t, x,ω) =1 +∞
2π
−∞ exp(−iβω)vβ (t, x, β )d β . (71)

Another important parameter is the relative co mputational It is known that the Fourier transform of the derivatives

accuracy ε1 . The algorithm stops if has the form

( )1  vn+1(x) − vn (x)  +  w n+1(x) − w n (x)  = ζvβ (t, x,ω) = (iω)vβ (t, x,ω),

∆ ζ2vβ (t, x,ω) = −ω2 vβ (t, x,ω).

∑=1  K | vn+1( xi ) − vn ( xi ) | + Similarly we can find the Fourier transforms of the
∆  i=0
unknowns vα (t, x, β ) , V (t, x, β ), p(t, x, β ) and their
K
∑+i=0 | w n+1( xi ) − w n ( xi ) | ≤ ε1. derivatives. Applying then the Fourier transform in form (71)
to each equation in (43')–(46') and each condition in (47)–(51)
Some results of computations are given in Fig. 3. and dropping the hats, we get the system

Alexander Blokhin et al.: On Stability of Curvilinear Shock Wave in a Viscous Gas 182

τ vβ + 1 ξvβ + iω p + b1vβ = χ{3iω( 1 ξvα + iωvβ −  R (β ) = LξVβ 1 +1 + L2Vβ  b1 + 3ω 2  +
L L    χ 
  χ  
1 (iωξvα + ξvβ 1 ξ 
−vα ) − L − L 2vβ ) − iωvα },  + LξWα ω (3 −1) − L2Wα ω (3 + 1) − L2ω ,
 χ

τ vα + 1 ξvα + 1 ξ p + a1vα − a2V = χ{3 1 ( 1 ξ 2vα 
L L L L +
LξWβ  1 + 1  b1 
  χ   χ 
  
+iωξvβ − ξvα ) + 3iωvβ + iω(iωvα + vβ − 1 ξvβ )},  (72) I (β ) = + L2Wβ + 3ω 2 +
+ iωvβ − vα ) +γ L 

τ p + 1 ξ p + 1 ( 1 ξvα d1 p + d2vα =  +LξVαω(1 − 3) + L2Vαω(3 + 1) + L2ω P,
L M2 L χ

= 2(γ − 1)χ{2a2 1 ξvα + 2(d1 − a2 )(iωvβ − vα ) + 
L 
R (α ) LξVα  1  L (ξP a2 L )
1 ξvα  =  χ3 + 1 + 3χ − +
+1d1 ( L + iωvβ − vα )},  
 

τV + 1 ξV − ( 1 ξvα + iωvβ − vα ) + d1(vα −V ) = 0.  L2  a1 + ω2 Vα + LξWβ ω  1  L2Wβ ω  1 
L L  3  χ  1 − 3 + 1 + 3 
    
with the conditions

ξvβ + (1 − 1 )Lvβ + iLωvα = 0,  I (α ) = LξWα 1  L (ξ − a2L ) +
χ  +1 + 3χ
  χ 3
 
3ξvα ε
+ (3 − 2)L(iωvβ − vα ) + L χˆ dˆ p = 0, (73) L2  a1 + ω2 Wα + LξVβ ω 1  L2Vβ ω  1 
 3  χ   −1 − 1 + 3 ,
  3  
V = vα ,  

1 R (V ) = ξVα − LωWβ − (1 + d1)LVα + d1L ,
p = − dˆ vα 


on the boundary x = 0 and I  (V ) = ξWα − LωVβ − (1 + d1)LWα + d1L ,

vβ = 0, (74)
vα = 
0  R ( p) = 2(γ − 1)χ (2a2 + 1d1) − 1  ξVα +
 M2 
on the boundary x = 1. 

Assuming vβ , vα , V and p to be complex-valued +  1 2 − d2 − 2(γ − 1)χ (2(d1 − a2 ) + 1d1) LVα +
M
functions in the form
+ω L  1 − 2(γ − 1)χ (2(d1 − a2 ) + 1d1 ) Wβ − γ d1LP,
vβ = Vβ + iWβ , vα = Vα + iWα , V =  + i , p = P + i, M

we substitute them into (72)–(74) and equate the 2

corresponding real and imaginary parts in the obtained I ( p) = 2(γ −1)χ (2a2 + 1d1 ) − 1  ξWα +
relations. Finally we derive M2 

L2 τVβ + R (β ) = ξ2Vβ , L2 τWβ + I (β) = ξ2Wβ , (75) 1 
χ χ  M2 
− d2 − 2(γ −1)χ (2(d1 − a2 ) + 1d1 ) LWα −

L2 τVα + R (α ) = ξ2Vα , ω L  1 − 2(γ −1)χ (2(d1 − a2 ) + 1d1)Vβ −γ d1L.
3χ  M2

L2 τWα + I  (α ) = ξ2Wα , (76) The boundary conditions for (75)–(78) at x = 0
3χ (77)
obtained from (73) have the form

Lτ  + ξ = R (V ), Lτ  + ξ = I  (V ), ξVβ + (1 − 1 )LVβ − LωWα = 0, 
χ 
 (79)
Lτ P + ξP = R ( p), Lτ  + ξ = I  ( p). (78) 1 0,
ξWβ + (1 − χ )LWβ + LωVα =
Here

183 International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186

3ξVα − 1L(ωWβ + Vα ) + L ε P = 0,  ψ n (x,ω) =ψ (n∆, x,ω) =ψ ,ψ n+1(x,ω) = Ψ , n = 1, 2,... ,
χˆ dˆ 
 (80) where ∆ is the step of the time greed.
0,
3ξWα + 1L(ωVβ − Wα ) + L ε  = (81) Approximating the derivatives τϕ (t, x,ω),
χˆ dˆ (82)
τψ (t, x,ω) in (85), (88) by expressions Φ − ϕ and
 = Vα ,  = Wα , (83) ∆
(84) Ψ −ψ
P = − d1ˆˆVα ,  = − 1 Wα .
d respectively, one obtains

∆
ξ2Φ = Φ +  (ϕ ),
Using (74), we find the boundary conditions (90)

Vβ = 0, Wβ = 0, where  = c1 ,  (ϕ ) = f1∆ − c1ϕ ,
∆∆
Vα = 0, Wα = 0

For (75), (76) at x = 1. ∆ ξΨ + Ψ = ∆ f2 + ψ . (91)
L L
6.2. The Numerical soluti on of B oundary-Value
Proble ms Let us discretize the relations (90), (91), (86), (87), (89)
with respect to the variable x . We introduce on the segment
Let the function ϕ be one of the unknowns Vβ , Wβ ,
[0,1] a uniform grid with the nodes xk = kh ,

Vα , Wα and the function f1 be the corresponding k = 1,..., K and the step h = 1 , where K is the
K
right-hand side R (β ) , IF (β ) , R (α ) or I  (α ) .
number of g reed nodes. Let Φ k , Ψ k , ϕk and ψk be th e
Then, the equations (75), (76) can be written in the general values of the unknowns Φ , Ψ , ϕ , ψ at the k th greed

form c1τϕ (t, x,ω) + f1 = ξ2ϕ (t, x,ω).

(85) node. Considering the boundary conditions (86), (87) on the

Each o f the boundary conditions (79), (80) can be n th time layer and replacing there the derivative ξϕ on

represented as (86) the boundaries x = 0 and x = 1 by its difference
analogues Φ 1 − Φ 0 and Φ K − Φ K−1 , one obtains the
ξϕ (t,0,ω) + c2ϕ (t,0,ω) + c3 = 0,
hh
and the conditions (83), (84) are

ϕ(t,1,ω) = 0. (87)

Remark 6.2. The coefficients c1 , c2 , c3 can be following boundary-value problem fo r equation (90):
ξ2Φ k = Φ k
written after elementary arith metical transformations of Φ 0 = A1Φ 1 + + (ϕ ) , k = 0,..., K,
B1, 
equations (75), (76) and the boundary conditions (79), (80).  (92)

Let the function ψ be one of the unknowns  ,  , Φ K − G1Φ K −1 = G2 , 

P ,  and the function f2 be the corresponding
1 c3h
right-hand side R (V ) , I  (V ) , R ( p) or I  ( p) . where A1 = 1 − c2h , B1 = 1− c2h , G1 = 1, G2 = 0 .

Then, equations (77), (78) can be written in the general form Remark 6.4. In section 5 we described an effective

Lτψ (t, x,ω) + ξψ (t, x,ω) = f2. (88) computational algorith m for searching solutions of

Each of the boundary conditions (81), (82) can be written boundary-value problems in fo rm (92) and based on the

as follows spline functions technique and the sweep method.

ψ = c4. (89) Replacing in equation (91) the derivative ξΨ at the

Remark 6.3. The coefficient c4 takes the values Vα , point xk by its difference analogue Ψ k − Ψ k−1 ,
h
Wα , − 1 Vα or − 1 Wα (see (81), (82)).
dˆ dˆ k = 1,..., K and taking into account (89), one gets

Our goal is finding appro ximate solutions of problem ∆ Ψ k − Ψ k−1 + Ψ k = ∆ ( f2 )k + ψ , 
(75)–(84) with certain init ial data. To this end we use the idea L h L 

of the method of lines and discretize the nonstationary k

equations (85), (88) with respect to the variable t . Introduce Ψ 0 = c4, 

the notations or

ϕ n ( x,ω) = ϕ (n∆, x,ω) = ϕ,ϕ n+1( x,ω) = Φ , n = 1, 2,... ,

Alexander Blokhin et al.: On Stability of Curvilinear Shock Wave in a Viscous Gas 184

{ }Ψ k 1 .
= ∆ + Lh ∆h( f2 )k + ∆Ψ k−1 + Lhψk  (93)

Ψ 0 = c4. 

Here ( f2 )k is the value of f2 at the point xk . Thus,
knowing the quantities ψk and ( f2 )k fro m the previous
time layer at each point xk (k = 0,..., K ) and Ψ 0 fro m

the current layer and using (93), we can find

Ψ k , ∀k = 0,..., K at the current time layer.

6.3. The Computati onal Scheme and the Results

O bt aine d

Thus, at each time layer we can find the values of the

unknowns Vβ , Wβ , Vα , Wα ,  ,  , P,  o f

problem (75)–(84) as solutions of the boundary-value
problems (92), (93). Then, using these values, we reco mpute

the right-hand sides R (β ) , IF (β ) , R (α ) , I  (α ) ,
R (V ) , I  (V ) , R ( p) , I  ( p) of equations (75)–(78 )

and pass to the next time layer (on the first layer the values of
the right-hand sides can be found fro m the initial data).

Choosing n large enough, we can finally find the

approximate stationary solution of problem (75)–(84) and,
hence, the approximate stationary solution of the original
problem (43 ' )–(46 '), (47)–(51) fo r large enough values of
time.

185 International Journal of Theoretical and M athematical Physics 2012, 2(6): 170-186

on Fig. 4 L = 10 ). But as you can see on the interval
x ∈[0.2,1] all the solutions are practically constant and
equal to zero. So we can suppose that for x > 0 all the

solutions are zero too.

ACKNOWLEDGEMENTS

The authors are indebted to prof. Yu.L.Trakhin in for the
help in the preparation of the manuscript of this paper, to prof.
D. L. Tkachev and Ph. D. R. S. Bushmanov.

REFERENCES

[1] P.D. Lax, Hyperbolic systems of conservation laws. II, Comm.
Pure. Appl. M ath., N 10, 1957, pp. 537-566.

[2] O.A. Oleinik, Discontinuous solutions of nonlinear
differential equations, Uspehi M at. Nauk, Vol. 12, 1957, pp.

3-73.

Figure 4. The functions [3] P.D. Lax, Nonlinear hyperbolic systems of conservation laws.
Nonlinear Problems, M adison: The University of Wisconsin
Vα (x),Wα (x),Vβ (x),Wβ (x), (x),  (x), P(x) and (x) Press, 1963, pp. 3-12.

obtained in the computations with the parameters [4] J. Glimm, Solutions in the large for non-linear hyperbolic
systems of equations, Comm. Pure Appl. M ath., N 18, 1965,
γ = 1, 4, Re = 150, M = 4, 3 = 2, L = 10, K = 1000 , pp. 697-715.

∆ = 0, 00001, ε1 = 10−4 , ω = 10 (see Table I) [5] J. Glimm, P.D. Lax, Decay of solutions of systems of
hyperbolic conservation laws, Bull. Amer. M ath. Soc., N 73,
The results of the numerical co mputations obtained with 1967, 105 p.
the help of the described algorith m are given in Fig. 4, where
one can see the stationary solutions for small perturbations [6] J.L. Johnson, J.A. Smoller, Global solutions for an Extended
Class of Hyperbolic Systems of Conservation Laws, Arch.
Vα , Wα , Vβ , Wβ ,  ,  , P and  . Thes e Rational M ech. Anal., N 32, 1969, pp. 169-189.

graphs show that the obtained values are sufficiently close to [7] C.C. Conley, J.A. Smoller, On the structure of
zero. This fact is the evidence of the stability of the shock magnetohydrodynamic shock waves, Comm. Pure Appl.
wave in a co mp ressible viscous gas with given parameters. M ath., N 27, 1974, pp. 367-375.

Remark 6.5. It should be noted that the computational [8] C.C. Conley, J.A. Smoller, On the structure of
problems considered in sections 5, 6 are rather co mplex due magnetohydrodynamic shock waves. II, J. M ath. Pures Appl.,
N 54(9), 1975, pp. 429-443.
to presence of large parameters L and Re∞ in the right
[9] N. Takaaki, J.A. Smoller, Solutions in the large for some
parts of equations (52) and (72). Moreover, in a nonlinear hyperbolic systems of conservation laws. Nonlinear
wave motion, Lectures in Appl. M ath., Vol. 15, Providence:
neighborhood of the line x = 0 we have the influence of Amer. M ath. Soc., 1974.

transitional zone of strong gradients and that is why near [10] B.L. Rozhdestvenskii and N.N. Yanenko, Systems of
quasilinear equations and their applications to gas dynamics,
x = 0 the solution has jumps and oscillations (see Fig. 4). Transl. M ath. M onog. 55, Providence: Amer. M ath. Soc., RI,
1983.
In this case the proposed computational algorithm for
solving boundary-value problems of second order prove it [11] S.P. D'yakov, On stability of shock waves, J. Exp. Teor. Phys.,
self to be very efficient and convenient. Indeed, the N (3)27, 1954, pp. 288-296; English transl. in Atomic Energy
nonstationary regularization that we used for searching the Research Establishment AERE Lib. trans., 1956.

stationary values of v and w and the application of spline [12] A.M . Blokhin, Estimation of energy integral of a mixed
problem for equations of gas dynamics with boundary
functions has a smoothing effect and allows to overcome the conditions on a shock wave, Sib. M ath. J., N (4)22, 1981, pp.
problems with ju mps and oscillations of solution during the 23-51; English transl. in Sib. M ath. J., N 22, 1981.
stabilization process.
[13] A.M . Blokhin, Uniqueness of the classical solution of a mixed
Remark 6.6. It is also important that we could not
obtain the stationary solution of the problem (72)-(74) for
really large values of L (in the experiment with results given