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Convergence of the pseudo-viscosity approximation for conservation laws

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Page 1: Convergence of the pseudo-viscosity approximation for conservation laws

Nonhew haiy& Tkwy, Metho& d Applicoti~~, Vol. 23, NO. 5, pp. 621-62&1994 Copyri&t 0 1994EbeviaSchceLtd

Prim!dinGr&Britain.AllIightcrcscmd 0362-s46x/9437.lkl+ .oo

CONVERGENCE OF THE PSEUDO-VISCOSITY APPROXIMATION FOR CONSERVATION LAWS

PIERANGELO M!mxTIt and ROBERTO NATAL@

t Dipartimento di Matematica Pura e Applicata. U&&t& degli Studi dell’Aquila, 14i7100 L’Aquiia; and $ Istituto per le Applicazioni de1 Calcolo, Viale de1 Policlinico 137, I-00167 R,ome. Italy

(Received 23 April 1993; received for publication 4 August 1993)

Key work and phrases: Conservation laws, shock waves, pseudo-viscosity, degenerate parabolic problems, gradient estimates.

1. INTRODUCTION

THB APPROXIMATION of shock waves involves both numerical and theoretical aspects and has been investigated by several mathematicians on both sides, see [l-3]. This paper is motivated from one side by the method of pseudo-viscosity introduced by von Neumann and Richtmyer [4] and on the other by the wish to study a good model problem of a gradient dependent viscosity.

In what follows we are interested in the following degenerate parabolic equation

c

f3,zP + aJ(U”) = &aJ?(aXUL), (X, 0 E R X R,,

ic”(x, 0) = u&), x E I?, (p,)

for E > 0, and in its limit as E 10

I

a,u + &J(U) = 0, (X,0 o R x R+, (P)

M-X, 0) = ll&), x E IR.

Let u. be a bounded measurable function and assume thatf is a Cz uniformly convex function; i.e.

(f) there exists c, > 0 such that f”(u) 2 c, > 0, for all u E IR. We recall that in this case the first result of convergence for the linear viscosity approximation, /3(d) = A, was proved in [5]. By contrast we will assume here that j3 is a C2 function with B(O) = 0 and such that

(8r) j?‘(A) > 0, /Y”(A) 5 0, for all Iz < 0, and /I’(A) = 0 for all d 1 0; (8J there exist some constants cr , c,, N > 0 and p 2 1 such that, for 1 < -N

c,]llp-‘A I /Y(A) I c21AIP--‘A.

A possible function B satisfying (8r) and (b2) is given by

(1.1)

which, as p = 2, is a well-known modification, for the scalar case, of ‘the pseudo-viscosity introduced in [4] (see also [2]).

621

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622 P. h4mxn and R. NATALINI

The arguments given in the body of the paper could also be used to deal with the case where assumption (8r) is replaced by

(Bi) there exists a constant N’ > 0 such that, for L < -N’, /3’(A) > 0, /3”(d) I 0, and 8’ = 0 otherwise.

Under the previous hypotheses it is not difficult to provide a heuristic explanation about the advantage of taking the approximation (P,) to study the problem (P). Let us consider a shock wave connecting two given constant states Us, on the left, and u,, on the right. Then, because of the convexity off and the Lax entropy condition, we must have uI > u,.

Let us now consider a travelling wave solution of (P,), namely a solution of the form

x - St ue(X, t) = u - ( > &

which approaches this shock wave as E + 0 The 24’ satisfies

where

tzx+, s = f(k) - f(4) u, - 4

and

lim u(r) = uf, [+-a0

{liy u(r) = u,. -DOD

For simplicity let u, = 1, u, = -1 and f(u) = iu”. Hence, for the usual linear viscosity, i.e. B(A) = A, the travelling wave is given by

u”(x) = -tanh f . 0

On the contrary as B is given by (1. l), with p = 2, the travelling wave takes the form

ue(x)= -sin(&)

i

-$7r<x5$7r,

-1 x> J

$r.

In this case the approximating wave spreads the shock only over an interval of the order of fi One of the most important advantages of using the term flA), given by (1. l), consists of the structure of the diffusion process which is driven, in this case, by (~/~x)(~~u/~~~~-‘(~ur/ax)). This operator will add just enough viscosity near shocks and as little as possible elsewhere. On the other hand, according to a suggestion of Rosenbhrth [2], there is no need to smear out the rarefaction waves, since they are increasing functions because of the entropy conditions. Other related numerical developments can be found in [2,6] and in the references therein, see also [7] for some rigorous results.

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Conservation laws 623

The mathematical difficulties arising in our analysis are mainly concerned with the existence of solutions to the problem (P,), which turns out .to be a problem of. mixed hyperbolic- parabolic type. Here the usual semigroup analysis does not show the real nature of the diffusive mechanism present in (P,); in particular with such an analysis it does not seem very easy to explain the nature of the L” bound (depending on E) on the derivative u,“.

This bound comes from the entropy conditions within the nondiffusive region and from the nonlinear diffusion elsewhere.

The solution to (P’) make sense both as “viscosity solutions” [8,9], and in the traditional weak sense. The results presented in this paper have some connections with the theory devel- oped in [lo], where different kinds of nonlinear viscosity have been considered.

We first consider the nondegenerate parabolic approximating problem

i

a,zP + a,f(zP) = &a,/3(a,ues*) + sa,‘zP”,

fP(X, 0) = z&x), vi,,)

for 6 > 0. In Section 2 we obtain the basic estimates for this problem, and in particular the sup norm

bounds for the gradient of ueV’, independently of 6. This is necessary to pass into the limit inside the nonlinear term /3(13,u”**). Section 3 deals with the convergence of ues6. First of all we let S + 0, and we get the existence for the problem (P,), then we let E + 0 and, by the methods of compensated compactness, we prove the convergence to an entropy solution of the scalar conservation law (P).

The results of the present paper can be partially extended to several space dimensions. This will be done, under more restrictive assumptions on the function 8, in a future paper [ 111, by using different methods.

2. BASIC ESTIMATES

The first lemma provides the energy and the sup norm bounds of the solutions to (P,,J.

LEMMA 2.1. For all T > 0 and u. E L” il L2, the solution uc** satisfies:

(i) IIu%Q 5 II~ollL-; 00 II~c%2 5 ll~ollL2;

Proof. The L” bound in (i) is an obvious consequence of the maximum principle. The inequalities (ii)- come from the energy identity

a,; jucs612 + a,[~o~*f’(s)Sdsj - a&e~*(&/3(a,ue’~) + 6a,u”*“))

+ axue~*(&/3(axue~6) + cmxUe*~) = 0. (2.1) By integrating in x and t on the strip 0 < t c T, one has

T

ss [ej?(u:*)u, + S&*1’] dxdt = ;

s (lu”‘*(x, O)]” - I@(+ T)j2) dx. (2.2)

0 R R

Since j?(A)d 2 0, the left-hand side of (2.2) is nonnegative, this immediately proves (ii)-( n

Page 4: Convergence of the pseudo-viscosity approximation for conservation laws

624 P. MARCATI and R. NATALINI

The next estimate regards the analogous of Oleinik’s entropy inequality [5]. It provides a unilateral bound for the gradient u?* which is due to the convexity off. It follows the full gradient bound for the expansion waves (in the nondiffusive region).

LEMMA 2.2. For all t > 0, E, 6 > 0, one has

lp(x, t) I f ,

where v is given in assumption (f).

(2.3)

Proof. Let us set u := U, (we shall drop the E and 6 labels for simplicity). We differentiate in x the equation in (PC,&) and obtain

v, + f’(u)v, + f”(U)V2 = (E/3(@ + 6v),,. (2.4)

By the maximum principle it follows at once that

for all c 1 l/v. H

Remark. If u. E W’,“’ then, by using as above the maximum principle, we easily get

u 5 ll~O,XlI~“. (2.5)

It is also easy to obtain the L’ type estimates both on the derivative U, and the modulus of continuity. In fact the following lemma holds.

LEMMA 2.3. Let w be a regular function and u any solution of the linear equation

a,o + &(f’(w)u) - McB’(w,) + &I = 0;

then it follows that

(2.6)

a4 + w’w)l~l) - uw’wx) + 6)bxlI s 0 in the sense of distributions. Consequently if u. E L’(lR)

s p6(X + h, t) - ueqx, t> 1 d.x I

s R luo(x + h) - u,(x)l dx

P

(2.7)

(24

for all h > 0, t 2 0. Similarly if 1~~ E W’~‘(lR)

IWC , oll,qn?) 5 Il~0,*ll&?). The proof is a quite standard argument (see for instance [12]).

(2.9)

Remark. By assuming u. e BY we also have

TV@“+, t)) 5 TV(z4,). (2.10)

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Consmvationlaws 625

Now we are going to establish a sup norm bound on u, and u, independent of 6, but depending, of course, on e. This estimate is necessary, since for fmed 8, we need to pass to the limit in the function /J(z@*), which, in general, is nonlinear.

PROPOSITION 2.1. Let ucB6 be the solution of (Q) and assume u. E IV’*“. Then there exists a constant M = M(c, T, ~~uo,,~~,-) such that

Iluj$- I M. (2.11)

Proof. Let us set ve*’ := ui*” (for brevity we will not repeat the indices E, s). The function v solves

Set

v, +f’(u)v, +f”(u)v2 = (&B’(V) + &lv,, + &/Y(V)V,z. (2.12)

w(X, t) := f(U(X, 0) - (E/?(V(X, 0) + 60(x, 0).

Hence, w is a solution to the linear equation

WC + f’owx = @PO4 + mm. In fact, multiply equation (Q) by f’(u) to obtain

fr + f’(u)fr = (e/W) + slf, + W’(v) + 4fY~)v2.

In the same way, we multiply (2.11) by w’(v) := (&B’(v) + ~5) to get

WC + f ‘(UMX = (EB’(~~ + 6)4%x

since (@?‘(v) + 6)@(v) = .@(v)~‘(v). so

II WIIP 5 llf Wo) - whx) + krllL~ *

(2.13)

Therefore, by using lemma 2.1(i), lemma 2.2 and the assumption (82), we obtain (2.11). n

The next result is concerned with the bound of us” independent of 6. This bound will be obtained by a straightforward application of the maximum principle to the equation differen- tiated in time.

PROPOSITION 2.2. Assume that E, 6 are small enough and u. E W2+‘(iR), therefore, it follows for anyT>O

where

M, := Il_f”(~“~*)u,“**ll.

Proof. Let us denote by w := u, , one has

w, + f ‘(NW, + fwwv = (&j?‘(V) + a)w, - @v, w,,

Page 6: Convergence of the pseudo-viscosity approximation for conservation laws

626 P.MARCATI and R. NATAUNI

Therefore, we immediately get a positive (resp. negative) supersolution (resp. subsolution) by solving the ODES

I $Y* f MY, = 09 f > 0,

c Y*(O) = *ll~o,tll~. The norm of u:,f in L” can be evaluated by using the equation, namely

ll~o,tlloo 4 Ilf<4l>,ll,- + 4s’<~o,J~o,nllL- + &J,&. n

3.CONVERGENCE

In this section we are concerned with the behaviour of the solutions to (P,,,) towards the solutions to (P,) as 6 + 0, and then we will pass into the limit when E + 0.

(a) Convergence us 6 + 0

As a consequence of the previous estimates, we claim the following result.

PROPOSITION 3.1. Let us assume a0 E I@“. It follows that there exists a subsequence (still denoted zP6) which converges, as 6 + 0, locally uniformly to a function ue; moreover, for all t > 0, [/~(u$~(s, t))] converges a.e. in x E R, to the function /3(ui(*, i)), and consequently ue is a weak solution to (P,),

Proof. Since (Up*“) and luf*‘) are uniformly bounded it follows that uePS + ue uniformly on compact sets as S + 0; moreover,

E/3(@) + su;** --* Es”

as 6 + 0, in L” weak-*. Hence,

ss la% + f (u”M, - &%A dx dt +

s u,$d.x = 0

v=oi

for all t$ E Ci(iI? x I?+). Thus, in the sense of distributions, it follows that

ut” + f (u”), = &Fx. (3.1)

As {@“], [f(zY*‘),) are bounded uniformly with respect to 6, we have that {/3(~@),) is uniformly bounded in L”. So, for a.e. t > 0

B(rP(~, 0) -b iw , Q

uniformly on compact sets, as 6 -+ 0. Taking into account that j3 is continuous and invertible, when restricted to the negative

semiaxes, it follows that, for a.e. f > 0

A(t) := Ix, jF(x, t) < 01

Page 7: Convergence of the pseudo-viscosity approximation for conservation laws

Conservation laws 627

is an open set. Therefore,

(3.2)

uniformly on compact sets in A(t). Hence,

u; = fl--‘(8’“) a.e. in a = lJ A(t), t>o

namely /F = B(ui) in a. Concerning the behaviour of j?(u:“) outside a we have to exploit the concavity of /.l. In fact

we have, for a.e. (x, t)

0 2 /3(u,“) r Eo/3(&) 1 /F.

Now, for (x, t) $ a, p(x, t) z 0 and then jl(ui) = 8” = 0. Consequently for a.e. (x, t) we have /?(u,“) = 8” and ue is a weak solution of (P,). n

The solution obtained in this way is clearly “admissible” in the sense of the “entropy inequalities”.

PROPOSITION 3.2. Under the previous assumptions, the previous solution ue of problem (P,) satisfies the following inequality

tit + 4x - E(tl’B)x s a in the sense of measure, for all convex functions q, where q’ = f’t~‘.

(3.3)

The proof is omitted.

(b) Convergence as E * 0

Let uc be a weak solution of (P,) obtained by the previous results. To use the methods of compensated compactness [13], we have to check that the equibounded sequence (zP] fulfills the following assumption

qr + qx E (compact set in Hz)

for all functions q, with q’ = f’q’. Because of the estimate in lemma 2.1(iii), the left-hand side of (3.3) belongs to a bounded set of measures. Therefore, we have only to show that

s(tl’(~“)/?(ui))~ E (compact set in W,&l*q)

for 1 < q s 2. One has, for all 4 E Cq, 4 2 0

where M := sup)~‘(u”)(. Let p be given by assumption (/3J and choose q := (p + 1)/p E (1,2]. Therefore, by the

energy inequality of lemma 2.l(iii)

IIP<~,>ll, 5 c,Pq.

Page 8: Convergence of the pseudo-viscosity approximation for conservation laws

628

Then in (3.4) we can

Consequently ~(tf’(fP)/3(B(t(,e)) --* 0 in W-‘*q as E + 0. We can now state the following theorem.

moRm+i 3.3. Let U, E W2+ and [u’) be a sequence of weak solutions of problem (P,) obtained as the limit of the solutions of problems (PQ. Then there exists a subsequence (still denoted ~8) which converges in Lf, to the (unique) entropy solution of (P).

REFERENCES

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-- Interscience, New York (1967). 3. SMOLLER J., Shock Waves and Reaction-Diffurion Equations. Springer, New York (1983). 4. NB~MNN J. VON & Rmmmm R. D., A method for the numerical calculation of hydrodynamical shocks, J. Appl.

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translation: Am. math. Sot. Transl. 26. 95-172 (1963). 6. ROACHB P. J., Computational Fluid Dynamics, 5th edition. Hermosa, Albuquerque (1982). 7. Uvua~ P. A.. Sur la r&solution num&iaue de l’&ruation (au/at) + u(&/&) - e(a/axHlau/axi(au/a = 0.

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