Viscous perturbations of isotropic solutions of the Keyfitz-Kranzer system

Appl. Math. Lett. Vol. 10, No. 1, pp. 51-55, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/97 $17.00 + 0.00

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V i s c o u s P e r t u r b a t i o n s of Isotropic Solutions of the Keyfitz-Kranzer S y s t e m F. HUBERT ENS Lyon, 46, all~e d'Italie 69364 Lyon C~dex 07, France

(Received and accepted March 1996) Communicated by D. Serre A b s t r a c t - - W e study large time behaviour of viscous perturbations of the Keyfitz-Kranzer system. We restrict our attention to isotropic solutions. Parabolic perturbations which enable the propagation of such solutions are computed. We describe the behaviour of their solutions for large time. Numerical perturbations are also studied, in the particular case of the Godunov scheme. We prove the existence of an explicit parabolic perturbation, whose dynamics for large time coincides with the Godunov scheme's one. K e y w o r d s - - H y p e r b o l i c system, Parabolic perturbation, Numerical perturbation, Asymptotic behaviour.

1. I N T R O D U C T I O N Let us consider a hyperbolic system of conservation laws:

Otu+Ox(f(u))=o, u(x, O) = a(x),

u • R ~,

x•R,

t>o,

x E R.

(1)

We a s s u m e t h a t the system is endowed by one linearly degenerate field a n d t h a t the initial c o n d i t i o n is periodic. We are i n t e r e s t e d in several p e r t u r b a t i o n s of this system. We deal, first of all, with parabolic perturbations

Otu~ + O x ( f ( u ~ ) ) : C o x ( B ( u ~)Oxu~), ~ ( x , o ) = a(x),

xeR,

t>O,

(2)

• • •,

where B is a n n x n m a t r i x , e a small p a r a m e t e r . We also s t u d y n u m e r i c a l p e r t u r b a t i o n s , involved b y a conservative scheme U ? "t-1 = U? - ,,~ (g (V?+l) - g ( U ? _ I ) ) , u ° = a(/~x),

j E Z, jCZ,

n E N,

(3)

where g denotes the n u m e r i c a l flux of the scheme, A x a n d A t the space scale a n d the time scale, A the ratio A t / A x . We s t u d y large t i m e b e h a v i o u r of such p e r t u r b a t i o n s ; in fact, we are more p a r t i c u l a r l y interested in t i m e t ~ r/-1, where r / = e in the parabolic case, or r / = A z in the n u m e r i c a l one. For time t << r/-1, the p e r t u r b a t i o n has a negligible effect on the solution, and we expect t h a t the Typeset by A.AdS-TEX 51

52

F. HUBERT

entropy dissipation associated to nonlinear field tends to shrink the amplitude of the so-called nonlinear waves to zero. For large time, we think that the solution depends deeply on the perturbation (see [1-3]). To point out such phenomenon, we study a 2 × 2 system, the Keyfitz-Kranzer system (see [4]), with an isotropic initial condition a~u + 0~(¢(lul)u) = 0,

u ~ R 2,

\sinwx]

'

x c ~,

t > 0,

x E R,

where r ° > 0 and w are given constants and ¢ is a convex function. Indeed, in this particular case, the dynamics for large time can be completely described for several perturbations. This system admits travelling wave solutions

U(X, t)

=r0 (cos

¢ (P) t) )

\sin(wx

¢ (r °) t) _ "

(5)

This solution remains isotropic for all time, that is Oxr - O, 0~@ =- w, if (r, 0) denote the polar coordinates of u. We want to study perturbations which enable the propagation of isotropic solutions. In a first time, we determine viscosity matrix B, for which the solution of the parabolic perturbation u~(z,0) --- u0(z),

z ~ R,

(6)

remains isotropic for all time. In a second time, we see that the Godunov scheme enables the propagation of such waves. In both cases, we give explicitly the solution and derive a large time behaviour. At least, we conclude that the solution of the Godunov scheme behaves, for large time, as the one of a certain parabolic perturbation, that we compute.

2. P A R A B O L I C

PERTURBATIONS

Let us determine the viscous perturbations of the Keyfitz-Kranzer system, which enable the propagation of isotropic solutions. We look for viscous matrix B, for which the system (6) admits travelling waves solutions

/ 0os

+ a°(t))

u~(x, t) = r~(t) ~ sin (wx + a~(t)) ] '

(7)

where r ~, a e are functions from IR+ to R. We remark that the hyperbolic system associated admits such solutions (B - 0)

u(x, t)

=p

t) ) ~,sin(wx

¢ (r °) t) _ "

(8)

However, we cannot expect the propagation of such a wave for any perturbations. PROPOSITION 2.1. The system (6) admits an isotropic solution for every isotropic data, if and only if there exists functions p: R + ~ R, q: ~+ ~-* ~, Z: R + ~-* ~2 L: R 2 ~-* ~2 such that

B ( u ) = [p(r)

Lq(~)

-q(r)]

p(r) J + Z(r) ® e; + Z(~) ~ e:,

(9)

where (r, O) denotes the polar coordinates ofu, e~ the vector (cos 0, sin 0), e~ the vector ( - s i n 0, cos @). PROOF. Let us look for necessary conditions. Rewrite the equation in polar coordinates ~ + (¢(~)r)~ = ~ [e~, (B(~)u~)~],

lee, (B(u)ux)x], we see that Otr, Or@ do not depend on the space variable x.

(10)

Viscous Perturbations

Let us note

53

V = -Oo(B(u)eo). It follows that v = p(,')e~ + q(,')~o,

and thus, the vector

[B(u)-(p(r)\q(r) - q ( r ) ) ] must only depend on r. Studying the matrix M such that Oo(Meo) = 0, we prove that B must have the form (9). Conversely, if B(u) takes the form (9), there exists an isotropic solution of (6) which satisfies

rt = -erw2p(r), Ot + ¢(r)w = -ewq(r).

(11)

Indeed, we can exhibit a solution to this system, as we will see further. But the problem (6) admits a unique global solution (we adapt the method of [5]), so that the solution of the system (6) is isotropic if and only if B(u) takes the form (9). | THEOREM 2.1. takes the form

For any viscous perturbation which satisfies (9), the solution of the system (6) u~(x,t) =r(et) (c°s(wx + a~(t)) ) sin (cox + ae(t)) '

(12)

with ), d r (=T--r(~-)w2p(r(7) ) d

~a

(t) = ¢(r(¢t)) +

(13)

¢coq(r(¢t)).

(14)

PROOF. Denote by r, a s the functions satisfying r ( ~ ) = r ~(~t),

aS(x,t) = O~(x,t) - cox. Obviously, r is solution of (13). The function a ~ is independent of x and verifies (14).

|

REMARK. These results have been extended to general periodic initial condition (see [1-3]). We find some function r0, 00, c such that (cos

(00 (x

ue(x,t) = ro(¢t) \ s i n ( 0 0 (x 3. T H E

GODUNOV

-

c(~t)t, ~t))

c(~t)t,ct)) J + 0(~). SCHEME

Let us introduce the solution of the Godunov scheme Uj~ (Uy c R 2, Vn E N, Vj ~ Z) given by the relation

u ; +1 = u ; -

vo

( f (w(o, u 2 , u ? + d ) - f (w(o, u L 1 , u ? ) ) ) ,

rO ( cos j A x ) \ sinjAx

where

(15) (16)

'

f(u) = ¢(]u])u and w(0, a, b) is the Riemann problem solution at the point (0, At): Otw 4- Oxf(w) = O,

W(X,O)

x E R,

j" a,

ifx < 0 ,

b,

ifx >0.

t > O,

54

F. HUBERT

We remark t h a t [U][ is independent of j, so t h a t the Riemann problem is reduced to a contact discontinuity

w(O,V?,U;n+l)= V ? , for all j, then w(O,U?,U?_{_I)= V?+l, for all j.

if ¢ (r °) > O, then if ¢ (r °) < O,

It is t h e n easy to see t h a t U1 takes the form

U1

T1 ( c ° s ( ° y j A x - J c ° ' l ) ) s i n ( w j A x + a 1)

'

and consequently by induction,

U~=rn(C°s(wjAx+an)) sin(wjax + an )

"

So, the G o d u n o v scheme is determined by the system

~n+~ = rn (1 + 2~ I¢nl (1 - ~ I¢nl) ( c o s ~ A x - 1)) v2 , r °,

given by (16)

(19) - A ¢ n sin w a x

a n+l = a n + a r c t a n

(18)

(1 -- ~ l e n d -[- ,~ I(~nl COSWAX'

a0 =0.

(20) (21)

Let us introduce the O D E system r'(t)=-axw2r(t)i¢(r(t))](1

- Al¢(r(t))l),

r ( O ) = r °, a'(t)=¢(r(~t)), a(o)

We cMl rn = r ( n A t ) , C F L condition

(23)

= o.

an = 8 ( n A t ) ,

(22)

Cn = r n - rn, ~n = a n - an. We suppose t h a t A satisfies the

;~sup I D f ( u ) l _< 1,

(24)

so t h a t we can prove the following. PROPOSITION 3.1. F o r a n y p o s i t i v e c o n s t a n t T , t h e r e e x i s t s a c o n s t a n t M s u c h t h a t : f o r all A x > O,

sup

nAtAx<_T

(Ic.I + 17.1) ~ M a x .

PROOF. An asymptotic development shows t h a t

rn+l = rn (1 - a ~ a x 2 ~ I¢"1 (1 - ~ ICnl)) + O ( a x 4 ) , r . + l = ~. (1 - a ~ a x 2 ~ ICnl (1 - ~ ICnl)) + o ( a z 4 ) , where O ( A x 4) is uniform with respect to n. We can deduce t h a t

(25) with an = ~

ICnl (I

- ~ lCnl) + ,'n(~I¢I(I - ~'I¢l))' (P'~),

(26)

Viscous Perturbations

55

where p~ E (r ~, r,~). Using the C F L condition, we prove t h a t

(27)

~ > ~ - 1 - KIe~l, so t h a t

I~+~1 ~ I~1

(i +

a ~ a x ~ (I + K I~1)) +

CAX4"

A n easy c o m p u t a t i o n shows t h a t , if n A t A x < T, t h e n

I~l ~ Max2.

(28)

T h e same m e t h o d gives ~n+l -- ?~n ~ a t sup ¢ ' ( r ) l e n [ 0 ( A x 3 ) , r<~ro

Using (28), the i n e q u a l i t y becomes

Inn[_ C n a z a.

1

We r e m a r k t h a t if we choose e = a x , p(r) = I¢(r)l(1 - Al¢(r)l), q - 0, Z -- 0, L -= 0, the s y s t e m (22),(23) is n o t h i n g else t h a t the system (13),(14); we can t h e n deduce i m m e d i a t e l y the following theorem. THEOREM 3.1. For any isotropic initial condition, the solution of the Godunov scheme Us for

the K e y f i t z - K r a n z e r s y s t e m is asymptotic for time t .., 1 / ( A x ) to the solution U ax of the viscous sys tern

ot~ + od¢(l~l)~) = a x G (p(I~I)G~),

(29)

where the function p is defined on R by

p(~) = I¢(r)l(i - ~1¢(,-),).

(30)

More precisely, for any constant T, there exists a constant M such that

s,p

jEZ,nAtAx<_T

IIu?- u ~x (jax,~at)ll _< M a x

(31)

REMARK. T h i s result has b e e n n u m e r i c a l l y verified for general initial d a t a (see [2]).

REFERENCES 1. F. Hubert and D. Serre, Dynamique lente-rapide pour des perturbations de syst6mes de lois de conservation, paraitre dans les CRAS, (1996). 2. F. Hubert, Dynamique lente-rapide pour des perturbations de syst6mes de lois de conservation, Th6se de Doctorat, ENS Lyon, (November 1995). 3. F. Hubert and D. Serre, Fast-slow dynamics for parabolic perturbations of conservation laws, Prdpublication de I'ENS Lyon 180 (1996). 4. B.L. Keyfitz and H.C. Kranzer, A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Rat. Mech. Anal. 72, 219-241 (1980). 5 D Hoff and J. Smoller, Solutions m the large for certain nonlinear parabolic systems, Ann. Inst. Henri Pomcard 2, 213-235 (1985). 6 F. Hubert and D. Serre, Slow dynamics of linear waves in nonlinear systems of conservation laws, In Proceedings of ICNEPDE, Beijing, (1993).