The cauchy problem for the equation describing acoustic waves in a medium with dispersion

57 6. IBRAGIMOV h.KH. and SHABAT A.B., On infinite Lie-Backlund algebras, Funkts. Analiz i Ego Prilozh., 14, 4, 79-80, (1980). 7. SOKOLOV V.V. and SHABAT A.B., Classification of integrable evolution equations, Math. Phys. Rev., 4, 221, 1984. 8. MIKHAILOV A.V., SHABAT A.B. and YANJLOV R.I., The symmetry approach to the classification of integrable equations. Complete lists of integrable systems, Uspekhi Matem. Nauk, 42, 4, 3-53, 1987. 9. Computer Algebra. Symbolic and Algebraic Computation, Computing Supplementum, 4, Springer Verlag, New York, 1982. 10. ZHARKOV A.YU. and SHVACHKA A.B., Investigation of the integrability of non-linear evolution equations using the analytical computation system REDUCE-2, Preprint Rll-83914, Dubna, OIYaI, 1983. 11, GERDT V.P., SHVACHKA A.B. and ZHARKOV A.YU., FORMINT - a program for the classification of integrable non-linear evolution equations, Comput. Phys. Comm., 34, 303-311, 1985; Computer algebra applications for classification of integrable non-linear evolution equations, J. Symbolic Comput., 1, 101-107, 1985. 12. ZAKHAROV V.E., MANAKOV S.V., NOVIKOV S.P. andPITAYEVSKI1 S.P.,Soliton theory: the inverse problem method, Nauka, Moscow, 1980. 13. GEL'FAND I.M., MANIN YU.1. and SHUBIN M.A., Poisson brackets and the kernel of the variational derivatives in the formal calculus of variations. Funkts. Analiz i Ego Prilozh., 10, 4, 30-34, 1976 14. BAHR K.A., Format 73 User's Manual, Darmstadt, GMD/IFV, 1973.

Translated

U.S.S.R. Comput.Uaths.Math.Phys.,Vo1.28,No.6,pp.57-64,1988 Printed in Great Britain

by Z.L.

0041-5553/88 $lO.OO+O.OC 01990 Pergamon Press plc

THE CAUCHY PROOBLEKFOR THE EQUATION DESCRIEING ACOUSTIC WAVES IN A KELIUf'i WITH DISPERSION*

V.V. VAP.LAMOV

The Cauchy problem is considered for the third order hyperbolic equation which describes the non-stationary wave processes in a medium with dispersion and absorption. The exact solution is found, and is estimated asymptotically with respect to the small parameter which characterizes the dispersionof the velocity of sound. For some years, there has been no slackening of interest in problems concerned with wave processes in media with dispersion and absorption. This is not only because such problems are of practical interest, but also because the relevant equations, which are often of a special kind, are worth studying theoretically. In the present paper we consider the Cauchy problem for the third-order hyperbolic equation that describes the propagation of non-stationary acoustic waves in a medium of the The problem was studied in /l/, where an integral form was obtained above-mentioned type. To solve the Cauchy for the fundamental solution, which contains modified Bessel functions. problem, the method of spherical means (see /2, 3/) was used. Here, we shall use a different approach, based on the construction of surface potentials, which was considered for the case of the wave equation e.g., in /4, 5/. We find a simpler expression than in /l/ for the We obtain an integral form for the fundamental solution, and study some of its properties. solution of the Cauchy problem, and time estimates for it, and the asymptotic expansion of the solution with respect to the small parameter that characterizes the dispersion of the This last result enables us to justify the "weak dispersion" approximation velocity of sound. of /6/. 1. Formulation of the problem. The propagation of linear acoustic waves absorption is given by the equation /7/

in a homogeneous

+$-cI.Au=O, ---__-___---*Zh.vychisl.Mat.mat.Fiz.,28.11,1685-1694,1988 USSRE&E-E

medium with dispersion

and

58 is the Laplace operator with respect to x, where u(z, 2) is the pressure, z=(zi, 52,zJ, A 7, co, and col signify the relaxation time and the limiting and the positive constants For convenience, we will introduce the dimensionless phase velocities of sound respectively. t=t/r, for which the previous notation is retained, and variables zr=zt/(c,z),i=l, 2, 3, and co'
2.

The fundamental

The general of a hyperbolic ficients. Since structed in /9/

solution.

Herglotz-Petrovskii relations, see /8/, hold for the fundamental solution coefa/at, d/dxi, i= 1, 2,...,N, with constant equation, homogeneous in Eq.(l.l) is not of this type, we use the function g(z,t), which was conand can be written as (Z.la)

(2.l.b)

K(p) which takes positive values for positive p; to isolate where we take the branch of [-1, -a]; /~~=(z,~fz~*-&~)", O(t)is Heaviside's this branch we make a cut along the interval function, and C+ is a simple closed contour which runs counter-ciockwise, lies in the halfand embraces the points p"-i and P'-a. plane RepGO, We show below that I(x, t) is the fundamental solution of the Cauchy Problem (l.l), We will (1.2). Let us study the function e(s,t) in the set G=((z, t) :.zER',IzI
Lemma 1.

We have

(2.2)

where ‘!a

r*(e)=

e==l-a,

-n
P&of.

Taking

as the contour

in it, we can write

C+in e(s,t)

[

COS$

)I . (1*cos+

(2.1) the circle as

Ip+al=e, and

putting

p=-a+eeiB

I

j exp((-or+ee'e)[t-(l+e-'e)'"ltl ]}de.

e&t)=+-

From this, after simple

reduction,

we obtain

(2.2).

I-a+ec"l~i, it follows from (2.3) that there is a number (2,t)'=G,,we have Iake/.3tkl
Note. Since that, with

(2.3

-r

N,>O such

et.3 0’ Another

expression

was obained

a.(z):” for (z,t)pG~. n-0 in /l/ for e(r,t):

e(x,Q==exp (alzj- qt

I{ z,(o)+

S [e(t-l~l)yr,(h!/)+hl,(hy)lI,(o(l-y’)’”)exp[e(t-l+/)y,2l}~y,

0

h=2[ae(t-lsl)/2]‘“, where r,(l) and I,(S) are the modified Bessel functions, o=E(t'-lZl*)"& and it was shown that 8(~,1) is a generalized function of slow growth which belongs to with respect to the variables 1.~1 and t, and is non-negative in G. Since this form C"(G) is quite laborious, it will not be used. e(%, t), considered in We shall require later some analytic properties of the function

59 the

set

a=(&

t) : IEl
C-0).

The function e(t&,lj and its derivatives Ler. 2. a"e(t&t)/at', k-1,2,... in the domain n, and there exist numbers do,d=const>o, such that we have M,-c&d’,

k=O, 1,.

are continuous

.. .

(2.4)

The continuity of the funCtions1be(1& t)/at*, k-O,&..., follows from (2.2). ProOf. 1*= TO obtain the bounds (2.4), we take as c+ in (2.1) the rectangle n+ with sides [-a, *ib, *ib], l,-[-a-ib, -a+ib], I,=[-ib, ib], where a>!. b>O, which contains p=-1 ,p=--a, as interior points. After simple reduction, we can write

elbl

1-al(a+p)

I) dP.

l+K(P)

Denote the exponential function on the right-hand side of (2.5) by F(p); we wil; obtain an estimate of the parameters a and b from the condition IF(p)l
K=(l+AFiB)“=K,WK,, K

Since

= R*:(l+A) 1.1 E 2

A = (~~s~~~bz

1’ ‘b

B=

eb (a-s)‘+bl

(2.6a)

’

R=[ (l+A)*+B*]“.

(2.6b)

jil
ab (a-s)*+b’

-KS N 1)

’

N=(l+K,)‘+K,z.

Since the power of the exponential function on the right-hand side of this last inequality is non-positive, we see that b22a. On the side 1, we put p=-aSip, -bGpG b. It then follows from the inequalities

IF(dIGeXP(-etlH[

i+

,~y$]~-

ab6a. It is easily shown that a similar inequality holds on the side that 1, for any positive valuesof a and b. Jp+al>a, IplG(a*+b*)“’ , We put a=6a and b=2a. Since, on the contour II+, we have we arrive at inequalities (2.4) by estimating and (1-K(p)(tlI
(2.7)

We shall say that the generalized function f(s, t) belongs to the class C'p'(a,b), O~P< m, with respect to the variable t, if, given any cp(z)=D(W, th e generalized function (f(z,t),cp(s)) belongs to the class C"'(%b). The next lemma shows that 8(+, 1) is a fundamental solution of the Cauchy problem. Lemma 3. The function a(.~.1)' belongs to the class variable t, and satisfies the following limit relations as

Cm[O, m) t++O:

with

respect

to the

(2.aJ)

60 (2.8c)

(2.8d) where 6(s) is the Dirac &-function and + denotes passage to the limit in the space o'(w'). We prove (2.8a) in a similar way to that used in /4/ when studying the fundamen; Proof. tal solution of the wave equation. Let us prove (2.8b). Let U(s,t) be the open sphere, radius t, centre the point 2, and let U, be the open sphere of the same radius and centre the origin. Let cp(z)=WIR'). By (2.1) and (2.71,

(2.9)

It follows from (2.2) that lime(t%,t)=i. I*+0

(2.10)

Bence, by Lemma 2, we see that the right-hand side of (2.9) tends to To prove (2.8c),we have

v,(O)

as

t+-l-0.

(

(2.11)

c(q.I)(&, VcpO%) )]d%+W,

where b(t)-0

as t-+0.

From (2.1) we have

where r+ is the positively oriented circle (p-l-a(=~.On evaluating the last integral, we obtain (2.12)

Notice #at (2.13)

bW)d%-0.

We obtain the required relation by passing to the limit as t+fO (2.12) and (2.13). On the basis of the above arguments, we can write (~(~,t).~(~))=~~[~(t%,t)rp(t%)

1%1-'+

in (2.11), and using

(2.14)

.z~(q,t)(~,v~(L%))+~(%,“(%.Vlp(f%)))]d%+ x(t), where x(t)+0 as t-+8. The operator acting on cp(t%) in the third term of the integrand is the second derivative with respect to the ditiction 5, and can be written as&V)'. The term itself can be rewritten as e(t%, t)(%/l%l, vcp(t%))+e(t%, t)(%T (%vv)vcp(t%))/l%I, and it can be seen that the integral over the domain u, of the first term of this expression tends to zero as t-+0 in accordance with (2.13). By (2.1),

61

Passing to the limit as (2.151, we see that

On evaluating

and using what has been

in Eq.U.14)

t-+0

the last integrals,

we

obtain

said,

and

(2.10),

(2.121,

(2.8d).

3. The Cauchy problem. Let us return to Problem (l.l), (1.2). Put IR+&=((z,t) :.z= RJ, t-=(0, a)}. a letter denotes closure of the set. Let u,(s),i-0, I,& be any generalized functions of D'(R'). We introduce potentials

(3.la)

(2, t) + ([u,(5)+~Ul(t)]X~r(f)},

V'"=1(2, where Lemma

the surface

u,t~)-~u.tr)-Au.t~)]] XW)}.

v(n)=8(s,t)*[(u,(l)+~[ V”4Y

The bar over

* denotes convolution, 3, we have

(3.l-b)

t)*[Uo(z)XG”(t)l, and

X

is

(3.k) the

direct product

of generalized

functions.

V')(z1 t) 1 i=O, 1, 2, belong to the class Lemma 4. The potentials C"[O, -) to the variable t, and satisfy the following initial conditions as t-+-IO:

P’(z, t) -+ 0, G

T

V”)(Z, t)* 0,

7

[ u,(z)-+)-AU&)

with respect

(3.2a)

(z,t)+O,

-a,(z)++ (z,t)

By

I ,

(2, t) + u,(r)++).

(3.2b) (3.2~)

(3.2d)

~(2,t)~~[Ug(=)+Au”tl)]. By the general

theory of Cauchy problems

(3.2f) /5, lo/, and Lemma 4, we have:

Theorem 1. The generalized solution of the Cauchy Problem (l.l), (1.2) exists and is ~=~'~1+~"'+ unique in the space D'(E+'). It can be written as a sum of surface potentials yc*, V"', i=o, 1, 2I are given by (3.1). ,, where Let us consider the classical Cauchy problem. V"'(X, Theorem 2. If i&(S)EC")(R'), U,(z)Ec"'(~') and u ~,(t)Ec"'(~~), then the potentials satisfy the initial conditions (3.2) in the Ccxp(~+‘), t), i=O, 1, 2, belong to the class classical sense, and for any point (x,t)ERJXIO,T],O
(3.3a)

t U=D(z,t), " )I

suplAuet

(3.3b)

(3.3c)

Proof.

By (2.1) and (3.la),

62 (3.4)

Here we make the changeofva$able u=tt, G-0. and belongs to the class C“'{R+‘). By (2.7) and (3.l.b), we can write

By Lemma 2, it now follows that (3.6) holds In accordance with (2.7) and (3.1~1,

By (3.4) and Lemma 2,

and that

Vu1

belongs

"(0, satisfies

to the class

(3.3a)

C(3)(R+').

(x-t%) d%+

-u.

Vu&-t%),&)+ e(t%,t) [(%,(*,

V)Vu.(l--l%)) f(~,Vs,(z-t%))])d%.

It is clear On estimating the last expression and using (2.4) , we see that (3.3~) holds. belongs to the class C’“‘(E+‘) . that V"' With regard to the initial conditions (3.21, it is clear from Lemma 4 that they are But we showed above that the satisfied in the sense of convergence in the space D'(R'L potentials V”‘(x, t), i=O, 1,2, belong to the class C@'(R+‘).Hence they also satisfy (3.2) in the classical sense. By Theorems 1 and 2, we have the following Theorem.

then the classicalsolution of Theorem 3. If UO(I)EC'")(R"),U,(l)EC("(RJ), u1(r)EC3'(RR'), Problem (l.l), (1.2) exists and can be written as (3.5)

where the functions e(z,t) is given by (2.2). This solution depends continuously on the initial data u,(s),i=o, 1,2, and on their derivatives up to and including the second order when (x,t)~lFX[0,Tj.

4. Asymptotic estimate of the solution. e=i-a=(c,'Notice that we can rewrite K(p) in (2.1) as K(p)-[[l+e/(ai-p)j"', where The parameter E characterizes the dispersion of the velocity of sound and, for many c,z)/cO'. actual media, is quite small, see /ll/. We therefore obtain below an asymptotic estimate with respect to E for the solution of our problem. Theorem 4. Let the functions Then, with O
u,(s),i=O, 1, 2,

satisfy

Ilu--nllc~‘+lR~x,o. .l,==ea, and on T.

of Theorem

3. (4.1)

u,(s),i=O,1,2, and on their derivatives up to and including e(r,t) The function iZ(r,t) is given by (3.5) if we replace

where the constant C depends on the second order, in it by

the conditions

63 o+,C.

w)=~

*

sexp@[t-JWI4 1)

(4.2)

dp_

a+P

o-1-

exp(-~I~l~2)exp[-a(~-lsl)lZ,(2[~(t-~s~)]’b),

where

Z,(x)

is the modified

Bessel

X(p)=1

G-0,

Proof.

ret

us

function of zero index,

show that, if

II

a% ---

aP

f---L

2(a+p) ’

and

p=-.44

2

O
k=O,

G&e’, at” II c~p+z,l

(4.3)

1,2,

where C,=Ch’T, C,OCa (note that (4.3) with k=O was obtained in /12/). We compare the functions e and e. For this, we transform in integrals (2.lb) and (4.2) to the rectangular contour II+, which was described in the proof of Lemma 2. On its horizontal sides I' we have H=1+A/2riB/2=~1TiK,, where A and B are given by (2.6). It can be shown that

K,-Hi

=

Kc-K,

=

BZ-(l+A)A’-A’/4 (K,+l+A/2)

(4.4&I)

(R+l+A+A*I2)

’

(4A+B’)B 2 (2’“+K,) (1-A+R)

are given by (2.6). The right-hand where R and K,_, The same bounds hold C,e' and C,e' respectively. We denote the powers of the exponential functions cD+iY and $i-iY respectively, where CD,Y, 5, by

sXp(@+iY)-exp(&+iY)=2

[

(4.4b)'

’

sides of (4.4) are upper-bounded by on the vertical sides l,,*. in th_e integrands of (2.1) and (4.2) and Y are real. We write the equation

@,+& &m-6 a'lYexp2 2

I

In the same way as when proving Lemma 2, we take a and b so large that we have the relationships exp[(Q+%)/2]S1 and exp(ifi)<1. On the other hand, it follows from estimates of (4.4) that ]sin[(Y-v)/2](9C,e'(x] and (sh[(~-;6)/2]liC,e'Ixl.Using the latter estimates and (4.5), we obtain inequalities (4.3). From (3.5) and (4.3), we obtain (4.1). It can be seen from (2.1) and (2.2), which describe the fundamental solution of our Cauchy problem, that a disturbance from an instantaneously acting point source 6(x)6(t) at 00 will be concentrated in a closed sphere, radius t, centre the point x=0. an instant Thus a forward wave front ]x]=t, is observed, which moves in the space with unit velocity. A disturbance is observed behind the forward front at all subsequent instants, so that the wave has no rear front. Analysis of the approximate relation (4.2) shows, however, that It there is nevertheless a pronounced maximum of the wave field at the level IxI=a’“t. occurs as a result of the influence of small terms of Eq.(l.l) and corresponds to a "second wave front", propagating with velocity a'". the threeNote that (3.5) is an analogue of Kirchhoff's formula. It shows that, in the wavesdescribed by Eq.tl.1) is similar to the propadimensional case, the propagationof gation obtained in the two-dimensional case from the wave equation. In conclusion, the author thanks A.V. Bitsadze and S.A..Gabov for useful discussions. REFERENCES 1. RENNO P., Sulla soluzione fondamentale di un operatore iperbolico della termochimica Rend. Accad. Naz. Sci. detta dei XL, Mem. di Mat., 4, 43-62, 1979-80. tridimensionale, 2. BERS L., et al., Partial Differential Equations, Wiley, New York, 1964. 3. JOHN F., Partial Differential Equations, Springer, Berlin, 1978. 4. VLADIMIROV V.S., 'Ihe equations of Mathematical Physics, Nauka, Moscow, 1976. 5. VLADIMIROV V.S., Generalized Functions in Mathematical Physics, Nauka, MOSCOW, 1979. 6. DUNIN S.Z., Wave propagation in weakly dispersive media, Prikl. Mekhan. i Tekhn. Fiz., 1, 138-141, 1986. 7. RUDENEO O.V. and SOLUYAN S.I., Theoretical Foundations of Non-linear Acoustics, Nauka, MOSCOW, 1975. 8. LEXE ZH., Hyperbolic Differential Equations, Nauka, Moscow, 1984.

64 9. VARLAMOV V.V., On the fundamental solution of the equation describing the propagation of longitudinal waves in dispersive media, Zh. vychisl. Mat. i mat. Fis., 21, 4, 629-633, 1987. 10. H&MANDBR L., Linear Partial Differential Operators, Springer, 1964. 11. NIKOLAEYVSKII V-N., Mechanics of Porous and Fissured Media, Nedra, Moscow, 1984. 12. VARLAMDV V.V., Asymptotic solution of initial-boundary value problem on acoustic wave propagation in a medium with relaxation, Differents. Ur-niya, 24, 5, 838-844, 1988.

Translated

U.S.S.R. Comput.Maths.Math.Phys Printed in Great Britain

.,Vo1.28,No.6,pp.64-75,1988

by D.E.B.

0041-5553/88 $lO.oO+O.oo 01990 Pergamon Press plc

KINETICALLY COORDINATED DIFFERENCE SCHEMES FOR MODELLING FLOWS OF A VISCOUS HEAT-CONDUCTINGGAS*

T.G. ELIZAROVA

and B.N. CHETVERYUSHKIN

The connection between the solutions obtained using kinetically-coordinated difference schemes (KCDS) and the solution of the Navier-Stokes equation for flows of a compressible viscous gas is analysed. It is shown that the KCDS model viscous flows and correspond to the Navier-Stokes equations with a Reynolds number that depends upon the spatial lattice step in the boundary layer. KCDS are constructed with a correction, and these model viscous flows with a real Reynolds number. The results of modelling a number of problenm, in which viscous and inviscid parts of the current have a significant interaction are briefly presented. Introduction. In /l-3/ a new approach to solving problems in gas dynamics was proposed based on kinetically-coordinated difference schemes (KCDS). KCDS differ from other algorithms for solving problems in gas dynamics as follows. As is well-known, the equations (Navier-Stokes or Euler) of gas dynamics are obtained by the well-known procedure of averaging from the Boltsmann equation for describing the one-particle distribution function /4/. In the difference approximation of these equations, we do not normally use the fact that these equations are themselves a corollary of a more-complicated transport equation. There is a small number of references /5-7/ in which macroscopic gas-dynamic parameters are determined by averaging the distribution function after solving the transport equation**). (**This remark only applies to a gaseous medium for which the Euler or Navier-Stokes equations are a valid description. To describe the behaviour of a rarefied gases, it is traditional to use different models of the transport equation). The computations carried out proved that such an approach is promising, and also showed the magnitude of the calculational difficulties standing in the way of its realization. These difficulties are, primarily, connected with solvingatransport equation that has greater dimensions than the equations of gas dynamics. In /l, 2/ a closed differential-difference equation system was obtained for finding gasdynamic parameters using simple kinetic models and, where necessary, the conceptofalocallyMaxwellian or Navier-Stokes distribution function. Consequently, it was possible /3/ to present these schemes as a result of the difference approximation of the Boltsmann equation. Such a treatment proved to be useful for further investigations in this direction and led to the construction of a new class of difference schemes in gas dynamics, these schemes being called kinetically-coordinated. Schemes of this class are constructed as follows: the difference scheme for the transport equation is written out, and this is then averaged with summator invariants. That is, unlike other algorithms for solving the equations of gas dynamics, in this case the difference discretisation process is carried out first, and then the averaging of the difference distribution function. Computations of inviscid hydrodynamic flows carried out using the KCIX proved that its use was promising, including its use for solving problems of the dynamics of a radiating gas The scheme has a good level of accuracy and its realization is relatively simple. The /8/. first attempts to use KCDS to calculate viscous currents taking account of the effects of the boundary layer arising on the solid-gas boundary also gave hopeful results /9, lo/. *Zh.vychisl.Mat.mat.Fiz.,28,11,1695-1710.1988