SIAM J. APPL. MATH.Vol. 52, No. 4, pp. 959-972, August 1992

() 1992 Society for Industrial and Applied Mathematics004

HOMOGENIZATION OF SCALAR CONSERVATION LAWSWITH OSCILLATORY FORCING TERMS*

WEINAN EtDedicated to Professor Kang Feng on occasion of his seventieth birthday.

Abstract. The homogenization of a scalar conservation law with a highly oscillatory forcingterm is studied. Effective equations are derived for the local averages of the oscillatory solutions,together with approximations that correctly represent the phase and the amplitude of the oscillations.A random choice method is also designed, which, as demonstrated by our numerical results, givesthe correct local averages without necessarily resolving the small scales. Finally, the homogenizationproblem with an additional small viscosity term is studied.

Key words, homogenization, oscillations, conservation laws, random choice methods

AMS(MOS) subject classifications. 35, 65, 76

1. Introduction. In this paper, we are concerned with the behavior and thenumerical computation of the solutions of a scalar conservation law with an oscillatoryforcing term

(1.1) i(x)u + f(ue)x -hHere e is considered to be a small parameter, and h(y) is a periodic function with meanzero. Although a homogenized equation for the weak limit (i.e., the local averages) ofthe sequence (ue}>0 can be readily derived using the results of Lions, Papanicolaou,and Varadhan on Hamilton-Jacobi equations [10], we will construct approximationsthat also give the correct phase and amplitude of the oscillations. In the contextof Hamilton-Jacobi equations, this amounts to the construction of correctors to theleading-order approximations obtained in [10]. We will also design a random choicemethod, which, as demonstrated by our numerical results, gives the correct localaverages of the solutions without resolving the small-scale oscillations. Finally, westudy the homogenization of (1.1) when a small amount of viscosity is added.

Equation (1.1) is an idealized model for a variety of interesting physical problems.When the convective part is replaced by the convection terms in the two-dimensionalincompressible Euler’s equation, the model describes what is commonly referred toas the Kolmogorov flow and is extensively studied in the literature as a model forunderstanding the inverse cascade process and the intermittency phenomena in tur-bulence [12]. Burgers equations with random forcing terms are used as the startingpoint for deriving turbulence models using renormalization group methods [8]. Ourknowledge on (1.1) can be beneficial to the study of these problems. One particularfeature of (1.1) is that the source term has large gradients. This presents both ana-lytical and numerical difficulties for studying the solutions. As we will show in 3, thestandard operator-splitting method for treating the source terms is very inefficient,and we should seek alternative ways to treat the source terms. There are numerousother problems, including some combustion models, where source terms with large

Received by the editors July 9, 1990; accepted for publication (in revised form) June 3, 1991.This research was supported by Office of Naval Research grant N00014-86-K-0691, Army ResearchOffice contract DAALO3-89-K-0039, and Air Force Office of Scientific Research contract AFOSR-90-0090.

Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012.

959

960 WEINAN E

gradients are encountered. We expect that a good understanding of (1.1) will alsoshed some light on these problems.

A closely related problem, the homogenization of the Hamilton-Jacobi equations

x Vve)__0 inRN (0 oo)v +H(,(1.2) v(x, O) vo(x) in RN

has already been studied by Lions, Papanicolaou, and Varadhan in [10]. Naturally,in (1.2), we assume that H(y,p) is periodic in y with period [0, 1] N. Under somenonlinearity conditions on the Hamiltonian H, it is proved that the solutions of (1.2),(v}, converge uniformly on compact sets to the solution of

(1.3) vt + H(Vv) 0, v(x, 0) v0(x),

where H (called the effective Hamiltonian) is given by the solution of the cell problem

(1.4) H(y,p + Vyw) A =/(p) in RN;

w is periodic in y with period [0, 1] N. It can be shown that, for any p E RN, thereexists a unique constant A (independent of y) such that (1.4) has a periodic viscositysolution. This constant A is defined to be the effective Hamiltonian at p.

There is a well-known simple relationship between Hamilton-Jacobi equationsand the one-dimensional scalar conservation laws. v is the viscosity solution of

+ f(v ) v(X), v (x, 0) uo(z)dz,

where V(y) f[ h(y)dy if and only if ue vxe is the entropy solution of (1.1). Asa consequence, we immediately get a homogenized equation for the weak limit of thesolutions of (1.1).

THEOREM 1.1. Let f(p) be the function defined in (1.4) with H(y,p) f(p)-Th n, O,

ue -- 2, weak * in LloCc (R R+),

where t is the unique entropy solution of

(1.6) fit + f(fi)x 0, fi(x, 0) uo(x).

Furthermore, a simple formula can be found for f in terms of f and V (see (2.8)).In this paper, we aim at understanding not just the behavior of the local averages,

but also the pointwise properties of ue. Instead of relying on the results of [10], we takea different and, in a sense, more conventional approach. Using a formal asymptoticexpansion, the problem becomes studying the large time behavior of the solutions ofthe cell problem

(1.7) u + f(u)y h(y).

We prove that, as T +, the solutions of (1.6) converge to steady state solutions.This allows us to construct pointwise approximations of u, which not only give thelocal averages, but also the phase and amplitude of the oscillations of ue. To geta closed equation for the local averages of u, we make the crucial observation thatalthough the periodic steady state solution of (1.7) is not uniquely determined by its

HOMOGENIZATION OF CONSERVATION LAWS 961

average (over the period), the average of the flux is. This allows us to successfullydeal with the closure problem. The above procedure gives an alternative derivationof (1.6). It does not directly give a rigorous proof of Theorem 1.1, since it is based onthe asymptotic expansion. We refer to [5] for a rigorous presentation of these ideas,along with a proof of the pointwise approximation result alluded to above.

Another interesting question to consider is the computation of these solutions.Traditional approach in numerical analysis assumes that waves of all scales are wellresolved and well represented by the computational grid. There are numerous physicalproblems for which it is very difficult, sometimes impossible, to meet this requirement.A typical example is given by the fully developed turbulent flow. Computing thedetails of such flows is beyond the ability of currently foreseeable computers. However,for such problems, we are often interested in their large-scale structures, rather thanthe details. Therefore it is natural to seek ways of computing the large-scale quantitieswithout resolving the small scales.

For many years, this problem has been approached using modeling. We seekequations that involve only the large scales and correctly describe the evolution ofthe large-scale quantities in the original problem. These equations are then used forcomputational purposes. An example can be found in Theorem 1.1, where (1.6) isan effective equation for the local averages of the highly oscillatory solutions of (1.1).While this approach has successfully solved a number of problems, it has also met greatdifficulties for others. First, it might be very difficult to obtain the correct modelingequations. This is the case for turbulent flows, where the notorious closure problemstill presents serious troubles for obtaining effective turbulent models. Second, solvingthese model equations numerically can be quite complicated. For example, to solve(1.6) numerically, we must know the solution to the nonlinear eigenvalue problem(1.4). In general, this can only be approached numerically. As a result, additionaliterations must be performed to compute the terms in the modeling equations.

An alternative approach was suggested in the pioneering work of Engquist [6],where he proposed to seek numerical methods based on the original differential equa-tions (not the model equations), that correctly capture the large-scale structures, evenif the small scales are not resolved on the computational grid. As was observed byEngquist, such methods should be free of the obvious numerical dissipations and dis-persions, since the numerical dissipations tend to damp out the small scales, and thenumerical dispersions tend to move the small scales to wrong locations and incorrectlyaccount for their effects on the large scales. Consequently, the traditional finite differ-ence and finite element methods are potentially not as good, compared to the spectraland particle methods, since the numerical dissipations and dispersions are inherent tothe former class of methods, but not the latter. Engquist went further to prove that,for Carleman’s equation with oscillatory initial data, a carefully formulated particlemethod does capture the overall large-scMe structures, even if the oscillations are notwell resolved on the computational grid. This work was extended to more general dis-crete Boltzmann type of equations in [7] and to the two-dimensional incompressibleEuler equation with oscillatory vorticity field in [3].

In all these problems, the velocity field that convects the particles or the vortex-blobs are rather smooth. That this does not hold for the present problem makesit closer to physical problems such as turbulence, since in a real turbulent flow wedo expect the velocity field to be oscillatory. However, it also makes the problemmuch more difficult. In 3 we present a carefully formulated random choice method(which is not eliminated on the bases of numerical dissipation and dispersion), which,

962 WEINAN E

as demonstrated by our numerical results, gives the correct local averages, even ifthe small scales are not resolved by the grid. Our formulation involves casting (1.1)into a 2 2 nonstrictly hyperbolic system, which itself has some interesting featuresregarding the solutions of the Riemann problems. We refer to [4] for another exampleof a similar kind, where the problem was approached using a very different method.

This paper is organized as follows. In the next section, we study the homoge-nization problem for (1.1). Then, in 3, we formulate our random choice method andpresent the numerical results. Finally, in 4, we study the homogenization problemwhen a small amount of viscosity is added to the right-hand side of (1.1). To empha-size the main ideas of our approach, we will omit the proofs of Theorems 2.1 and 4.1and refer the interested reader to [1], [2] for the missing details.

2. The homogenization problem. We begin with a formal asymptotic argu-ment based on the following ansatz"

u + Ul +.

where w(x, t, y, T), Ul (X, t, y, T) are assumed to be periodic in y with period I [0, 1]and have sublinear growth in T. Following standard procedures in multiple-scaleanalysis, namely substituting (2.1) into (1.1), and equating the coefficients for powersof , we obtain

(2.2) w + f(w)y h(y),

(2.3) wt +u + f’(w)(wx + uy) + f"(w)uwy O.

Equation (2.2) describes the evolution of the oscillatory component of ue. Using stan-dard terminologies in homogenization theory, we will refer to it as the "cell problem."

2.1. Behavior of the solutions to the cell problem. Since T tie and issmall, we will only be interested in the large time behavior of its solutions. For this,we have the following result.

THEOREM 2.1. Assume that f is C3, strictly convex, and that f(u) -- +oc as[u --, +o. Let v(y, -) be the solution of (2.2) with initial data v(y, O) a(y), wherea(y) is a periodic function with period I and has locally bounded variations. Then, asT +cx3, v(y, T) converges in L(I) to a steady state solution of (2.2).

The proof of Theorem 2.1 is quite involved, since system (2.2) exhibits some ofthe features of a 2 2 system with one linearly degenerate field. Two types of wavesare present in the solutions of (2.2): the shocks and rarefaction waves of the scalarconservation law u+f(u) O, and the standing waves ofu h(y). The interactionof these waves drives the solution to a steady state.

From Theorem 2.1, we know that when is small, w(x, t, y, T) is very well approx-imated by a function that is independent of T. This means that, to the leading-orderapproximation, the spatial oscillations in the forcing terms of (1.1) do not excite tem-poral oscillations. This is not true if the flux f is linear, as can be seen from thesimple example ut + ux (1/)sin(x/), uo(x) 0. The solution of this equation isu(x, t) cos(x/e) cos((x t)/). This is another evidence that the nonlinearity ofthe flux greatly influences the behavior of the solutions.

The steady state solutions of (2.2) are determined by f(U)y h(y) plus theentropy condition

lim v(y, T) > lim v(y, T)

HOMOGENIZATION OF CONSERVATION LAWS 963

for Y0 E I, T > 0. Integrating the above equation once, we get that

(2.4) /(u) c,where Y(y) f h(s)ds and C is a constant.

Equation (2.4) suggests the study of the phase plane portrait of the Hamiltoniansystem with Hamiltonian H(y, u) f(u)- Y(y). It is easy to see that when f satisfiesthe conditions in Theorem 2.1, there are two types of qualitatively different orbits:bounded orbits that represent periodic motions, and the orbits that extend to infinityin y. Since the Hamiltonian is constant along every orbit, each of the latter type oforbits gives a (smooth) steady state solution. Furthermore, we observe that

(1) The periodic orbits do not correspond to any steady state solutions of (2.2),since these orbits cannot be graphs of single-valued functions;

(2) Discontinuous solutions can occur only at the separatrices. This is a conse-quence of periodicity and the entropy condition.

Since (w} f w(y, T)dy is a conserved quantity, it is natural to ask whetherthe steady state solutions of (2.2) are uniquely determined by their averages. Thefollowing result provides a partial answer to this question.

THEOREM 2.2. Assume that f(u) satisfies the conditions of Theorem 2.1. Assumealso that Y(y) attains different values at different local minima on [0, 1). Then thesteady state solutions of (2.2) are uniquely determined by their averages.

Proof. Let ul(y), u2(y) be two steady state solutions of (2.2) such that ul u2.First, let us consider the case when both ul and u2 are continuous. If (ul} (u2}, thenthe graphs of Ul and u2 intersect at at least one point, say (y*, u*). This point mustbe a saddle point for the Hamiltonian H(y, u), and ul, u2 must lie on the separatricescoming out of (y*, u*).

After a coordinate transformation, we can assume that y* 0. The conditions on

f and Y ensure that there are precisely two orbits (y, 1 (y)) and (y, 2(y)) connecting(0, u*) and (1, u*), and neither intersects any other orbits on (0,1). Without loss ofgenerality, we can assume that 1 > 2 on (0,1) and 1(0) 2(0) 1(1)2(1) u*. Since (1} > (2}, it is clear that (ul} (u2}, unless ul u2 i, 1or 2.

Suppose now that Ul contains a shock. (The case when u2 contains shocks canbe treated similarly.) From the entropy condition and the periodicity, u must havethe following form:

1(Y), 0 _< y < C1,Ul( ) < < 1,

where 0 < C1 < 1; 1 and 2 were defined above. Since the graph of u2 cannot fallinto the region bounded by 1 and 2 (this is a consequence of the strict convexityof f), either u2 has the same form as ul with a different value C2 E (0, 1), or u2 iscontinuous. In either case, (Ul} (u}. This completes the proof of the theorem.

When the conditions in Theorem 2.2 are violated, there can be infinitely manysteady state solutions, all having the same averages. Consider a case when V hastwo local minima Yl, Y2 in [0,1) and V(yl) V(y2), 0 <_ yl < y2 < 1. In this case,we have that 1(yl) 2(yl) )l(y2) 2(y2) 1(yl + 1) 2(yl d- 1) u*,where (y, 1(y)) and (y, 2(y)) are the separatrices connecting (yl, u*) and (y2, u*),(y2, u*) and (yl + 1, u*). The entropy solutions can have shocks in the interval (Yl, Y2)or (Y2, Yl + 1). The shock locations can change continuously without changing theaverage of the solution. Explicit examples are given in [10].

964 WEINAN E

We remark that, although there can be infinitely many steady state solutions of(2.2) with the same averages, they all give rise to the same value of the HamiltonianH(y, u) f(u)- V(y). We formulate this as the following theorem.

THEOREM 2.3. Let ul and u2 be two steady state solutions of (2.2) such that(ul) (u2). Then H(y, ul) U(y, u2).

Proof. This is more or less obvious from the proofs of Theorems 2.1 and 2.2. Ifu : u2, then ul and u2 lie on the separatrices. These separatrices must span thewhole period I. Obviously, the Hamiltonian is the same on all such separatrices.

This result will be crucial for closing the hierarchy of equations given by theasymptotics.

2.2. The homogenized equation and the behavior of the homogenizedflux. Now let us return to the set of equations (2.2) and (2.3). Denote by (.) theoperation of taking the average of a periodic function over its period. As we haveremarked earlier, according to (2.2), (w) does not depend on T. Therefore we candenote it by fi(x, t). Averaging both sides of (2.3) with respect to y, we get that

(2.5) tt -- (Ui)- -- (f(w))x / (f’(w)Uy / f"(W)UWy) O.

The last term on the left-hand side of (2.5) ((ft(w)ui)y) 0. Next, we integrate(2.5) with respect to T from 0 to T, divide both sides by T, and let T - +x). Usingthe sublinear growth condition (1/T)(u(x, t,., T)) --. O, we get that

(2.6) t + (f(w)}x -O,

where (f(w)) limT_(1/T) ff(f())d.For fixed x and t, w(x, t, y, T) goes to a steady state, denoted by (x, t, y), as

T -- +cx. Therefore (f(w)} (f((v)}. Note that ((x, t, .)} ft.

To get a closed equation for , we must represent (f()} in terms of . This ispossible due to Theorem 2.3. Even though we cannot uniquely determine in termsof fi, the average of the flux is determined according to Theorem 2.3. In fact, we havethat (f()} U + (V). Therefore we conclude that (f()} is a well-defined functionof (z} ft. We write this as (f()) f(fi). This is a unique property of the flux fnot shared by other nonlinear functions of u. Fortunately, to close the hierarchy ofequations at this level, we only need the average of the flux f, and this gives us thefollowing homogenized equation"

(2.7) fi + f(fi)x -0.

Assume that f satisfies the conditions of Theorem 2.1. We normalize f(u) andV(y) such that f(0) 0, f >_ 0, and min V(y) 0. We will use f- to designate thenonnegative branch of the inverse function of f. It is easy to see that f is given by asimple formula

(2.8) ](p)_ ( O, if Ipl <- (f-(V(y))),

In Fig. 1 we display the homogenized flux with f(u) u2/2, h(y) 2sin2ryand compare it to the original flux. The fiat piece on the homogenized flux as wasdiscussed in [10] is clearly shown. Another important piece of information revealedby Fig. 1 is that, away from the fiat piece, the homogenized flux is very close to theoriginal one.

HOMOGENIZATION OF CONSERVATION LAWS 965

4.5

4

3.5

2.5

1.5

0.5

00 0.5 1.5 2 2.5

FIG. 1. Comparison of the original flux/(u) u2/2 and the homogenized flux f. Note thatis flat in a neighborhood of the origin.

2.3. Pointwise approximation of the oscillatory solution. To get a point-wise approximation of u, we must study the full function w(x, t,y, T) in (2.1). Aswe showed earlier, for fixed x, t, w(x, t, y, T) goes to (x, t, y) as T - +Cx, which is asteady state solution of (2.2), and ((x, t, Y)/- fi(x, t). Therefore we expect to have,as e - 0, that

(2.9) lu (x, t, x)l o,

where (x, t, y) satisfies

(2.10) f((x, t, y)) V(y) + f(fi(x, t))

and fi(x, t) is the solution of (1.6) with initial condition fi(x, 0) -so(x).If f and V satisfy the conditions of Theorem 2.2, then there is a unique solution

of (2.10), which, according to (2.9), gives a pointwise approximation of the oscillatorysolutions of (1.1).. The rigorous proof of this result is presented in [5]. Here we willshow our numerical results on a comparison between u and . Below we will reportour numerical results for two sets of data. In both cases, we take f(u) u2/2, h(y)2 sin 2ry, and I-J0, 1] as our computational domain with periodic boundary conditionenforced. The exact solution u was computed using the Engquist-Osher schemewith very high resolution (50 points per wave). In the first case, we take so(x)5 + 3 cos 2rx, e 0.0333, and t 2. The results are shown in Figs. 2(a) and 2(b). InFig. 2(a) we display the superposition of ue and (v(x, t, x/e) computed from inverting(2.10). Their difference is shown in Fig. 2(b). We see from these figures that, asidefrom a small neighborhood of the shock, u and are indeed very close. In thisparticular example, since u0 lies in the region where f is very close to f, fi is computedby simply using the flux f. In the second example, we take so(x) 0.7, for x < 0.5and so(x) -0.7, for x > 0.5; e 0.04, t 2. The amplitude of this initial data

966 WEINAN E

5.3

5.2

5.1

4.70 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIG. 2(a). Superposition of u and Co, with initial data uo(x) 5 -t- 3cos27rx; 2, e 0.0333.

0.005

-O.Ol

-0.015

-0.02

-0"0250 0.1 0.2 0:3 0:4 0:5 0:6 0:7 0:8" 0:"

FIG. 2(b). The error between u and Co. Other parameters are the same as in Fig. 2(a).

is small enough such that the homogenized flux f is flat in the range of u0, and thehomogenized equation for the local averages is just fit 0. The numerical resultsfor this initial data are presented in Figs. 3(a) and 3(b). Similar conclusions can bedrawn from these figures as in the previous example. Furthermore, Fig. 3(b) clearlyshows the formation of cusps in the microstructures. We point out that, in this case,the leading-order approximation z is independent of t. This can be seen easily from(2.10), since fi does not depend on t. More numerical results can be found in [1].

HOMOGENIZATION OF CONSERVATION LAWS 967

1.5

0.5

"1"50 0.1 0.2 0.3 0.4 0.5 0:6 0:7 0.8 0.9

FIG. 3(a). Superposition of u and , with initial data uo(x) 0.7 for z > 0.5, and uo(x)--0.7 for x < 0.5 2, 0.04.

0.5

-0.5

-1.5

-20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIG. 3(b). The error between u and Cv. Other parameters are the same as in Fig. 3(a).

3. A random choice method. In this section, we present a numerical methodfor solving (1.1) that captures the correct local averages, even if the high-frequencyoscillations are not well resolved by the computational grid. We emphasize that themethod is based on directly solving (1.1), not the homogenized equation (1.6).

The conventional shock-capturing methods, including the Lax-Friedrichs and Go-dunov type of methods, introduce artificial viscosity to guarantee a monotone tran-sition in the shock layer. When the grid size is substantially larger than the high-

968 WEINAN E

frequency wavelength, the high-frequency oscillations are effectively smeared out inthe initial step. Since these oscillations have an O(1) contribution to the large-scaleaverages at later times, it is clear that such methods cannot achieve what we desired.Another disadvantage of the usual finite difference methods is that they are inevitablydispersive, although the amount of the numerical dispersion can be reduced by raisingthe order of the methods. Due to the numerical dispersions, the high-frequency wavesmove at a wrong speed. This is another reason that such methods will not correctlycount for the effect of the high-frequency oscillations to the large scales. On the otherhand, the random choice method (see [9]) does not have the above-mentioned nu-merical dissipations and dispersions. No averaging process is introduced, cell-to-cellcommunications are needed only to get the exact solutions to the Riemann problems,and all waves move at the right speed; therefore it is not eliminated on the basis ofnumerical dissipations and dispersions. As we show in this section, a carefully formu-lated random choice method does have the feature specified at the beginning of thissection.

For simplicity, we consider the case when f(u) u2/2. Let (u, v) be the solutionof

(3.1) u + (f(u) + ve) O, v O,

with initial condition ue(x, O) Uo(X), v(x, O) -V(x/s), where V satisfies V’(y)h(y). Note that (3.1) is not a strictly hyperbolic system: the Jacobian matrix is notdiagonalizable at u 0.

To solve its Riemann problem, we need an appropriate entropy condition. Recallthat Lax’s shock condition requires that at each point on the shock, three character-istics enter the shock and one leaves the shock. This was generalized by Keyfitz andKranzer to accommodate the following situation: one characteristic enters the shock,one leaves the shock, and the other two are parallel to the shock. For our problem,the entropy condition must be further generalized to allow the incoming or outgoingcharacteristics to become parallel to the shock. We state this below.

Generalized shock condition. A jump is said to satisfy the generalized shockcondition if only one characteristic leaves the jump.

With this, we can solve the Riemann problem associated with (3.1), with theinitial condition

U(x, o) o) o)) { x<O,x>O.

Obviously, we have that

v(x, t) x < 0,Vr, x >0.

The jump of v at x 0 will introduce a jump in u. Let u+ limx__.0+u,u_ limx_0- u, then the Rankine-Hugonoit condition at x 0 is

u2

_2 +vt +v,..

It is easy to see that Lax’s shock condition implies that u+. u_ >_ 0.In the following, instead of giving the details of the derivation, we will simply give

the solution to the Riemann problem. The reader can readily check that this is the

HOMOGENIZATION OF CONSERVATION LAWS 969

5.3

5.2

5.1

4.9

4.8-

4.70 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIG. 4(a). Numerical solution of the random choice method, with initial data uo(x) 5 +3cos27rx; 2, e 0.00013798473, /kx 0.005, At 0.0005.

only self-similar solution satisfying the generalized shock condition. Here, by a shockwave (or a rarefaction wave), we mean the shock wave (or rarefaction wave) solutionof the Burgers equation ut + (u2/2)x 0.

Case 1. uz > O, ur < O. In this case, ifu/2+vz < u/2 +vr, then u+u,u_ -x/(u2/2 + v- vz) 1/2. ut and u_ are connected by a shock wave. Ifu/2 + vt > u2/2 + v, then u_ ut, u+ x/ (u/2 + vz vr) /2. u+ and ur are

2/2+v thenu ut for x < 0, andconnected by a shock wave. If u/2 + vt uu- u for x > 0. This is an overcompressive wave [11].

Case 2. ut >_ 0, u > 0. In this case, if u/2 + vz vr >_ 0, then u_ ut,u+x/ (u/2 + vz vr) /2. u+ and u are connected by a shock wave or a rarefactionwave, as in the case of Burgers equation. If u/2 + vt v < 0, then u+ 0, u_-x/(v- vt) /2. u+ and ur are connected by a rarefaction wave. uz and u_ areconnected by a shock wave.

2/2+v vt_>0, thenu+ u<0, u_-Case 3. ut <_ 0, u < 0. In this case, if u-x/ (u2/2 + vr-vz) /2. ut and u_ are connected by a shock or rarefaction wave, as in

2/2/v vz<0, thenu -0, u+ x/(vz v)1/the case of Burgers equation. If uu+ and u are connected by a shock wave. ut and u_ are connected by a rarefactionwave.

Case 4. uz <_ O, u,. > O. In this case, if vz < v, then u_ 0, u+ x/ (vz-v) 1/2,uz and u_ are connected by a rarefaction wave, u+ and u are connected by a shockwave or a rarefaction wave. If vz _< v, then u+ O, u_ -x/ (vr- vz) /2. u+and ur are connected by a rarefaction wave, uz and u_ are connected by a shock orrarefaction wave.

We next come to the numerical results of this method. Again, we take

U2

:f (u) --, h(y) 2 sin 2try.

In Figs. 4(a) and 4(b) we display the numerical solution of this random choice method

970 WEINAN E

5.3!

5.2

5.1

5

4.9

4.8

4.70 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIG. 4(b). Comparison of the local averages of the above numerical solution and the solutionto the homogenized equation. The numerical parameters are the same as in Fig. 4(a).

with initial data no(x) =5 + 3cos27rx, the same as the one used in Figs. 2(a)and 2(b). The numerical parameters are /x 0.005,/t 0.0005, t 2, and6 0.00013798473. Figure 4(a) displays the numerical solution, while, in Fig. 4(b),we compare the local averages of the numerical solution to the solution of the homog-enized equation. Similar results for the initial condition used in Figs. 3(a) and 3(b)can be found in Figs. 5(a) and 5(b).

We might naively treat the forcing term by the splitting method; i.e., in eachtimestep, we first solve the Burgers equations ut + (u2/2)z 0 and then solve an

ordinary differential equation ut lh(x/6) to take care of the forcing term. Moresophisticated splitting schemes can be constructed to achieve second-order accuracy.This is disastrous if we take the grid size /x to be much larger than 6. We musttake very small timesteps /t < C6 to prevent the numerical solution from blowingup immediately. Even then, the numerical solution blows up anyway, only later.

We might also try to improve the random choice method such that, at eachtimestep, we update the values of v by evaluating V(x/6) at the newly chosen randompoints. This destroys the overall conservation property of the random choice method.When used with /x >> 6, although the numerical solution remains bounded, thelocal averages have O(1) errors.

4. The effect of viscosity. In real physical situations, a small amount of vis-cosity is often present. Roughly speaking, the viscosity will try to smooth out theoscillations generated by the oscillatory forcing. The competing effects of a smallamount of viscosity and a strongly oscillatory forcing is the subject of this section.

Consider the following model:

(4.1) u + f(uS)x e" s us+ (x, O) uo(x).

h(y), U(y) were defined in 1, except that we normalize U such that (V) 0. Then

HOMOGENIZATION OF CONSERVATION LAWS 971

1.5

0.5

-0.5

-1-

"1"50 0:1 0:2’ 0:3 0:4 0:5 0:6 0:7 0:8 0:9

FIG. 5(a). Numerical solution of the random choice method, with initial data uo(x) 0.7 forx > 0.5 and uo(x) -0.7 for x < 0.5; 2, e 0.000397643973, Ax 0.005, At 0.0005.

81\/’xx/’X A A A A/’X/’Vx/NA A0.6 t kl J J V v k,lJll0.4 II0.2

-0.2

-0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIG. 5(b). Comparison of the local averages of the above numerical solution and the solutionto the homogenized equation. The numerical parameters are the same as in Fig. 5(a).

we have the following theorem.

THEOREM 4.1. Let T < +oo, 1 < r < +oo. Assume that f is convex, and+oo

(4.2) ue --+ u weakly in L[oc(R x [0, T]),

972 WEINAN E

where u is the solution of

(4.3) ut + ](u)x O, u(x, O) no(x),

and fa is determined by the following conditions:(1)/f 0 < a < 1, then ](p) f(p), p e R1. Furthermore, if f is strictly convex,

then u converges strongly to u in Lloc(Rr [0, T]).(2) If ( 1, then, for any p E R1, there is a unique constant ) such that the

following problem has a periodic (viscosity) solution with period [0, 1]

(4.4) f(p + vH) vHH + V(y) + ,and f(p) .

(3) If a > 1, then ],(p) is the same as f(p) defined in 1, f(p) f(p), p e R1.The theorem is proved by going to the level of the Hamilton-Jacobi equations.

The proof involves deriving the necessary estimates for compactness, identifying theproperties of the limiting semigroup, using the result of Lions to conclude that it isthe semigroup given by a Hamilton-Jacobi equation, and identifying the Hamiltonianof this namilton-Jacobi equation. The details are given in [1], [2].

Acknowledgments. I thank my advisor, Professor Engquist, for his guidanceduring the preparation of this paper. I am also grateful to Drs. P. L. Lions, S. Osher,and C. W. Shu for helpful discussions.

REFERENCES

[1] W. E, Homogenization and numerical methods for hyperbolic conservation laws with oscillatorydata, Ph.D. thesis, University of California, Los Angeles, 1989.

[2] , Homogenization of scalar conservation laws with oscillatory forcing terms--expandedversion, unpublished manuscript.

[3] W. E AND T. HOU, Homogenization and convergence of the vortex method for 2-D Eu-let equations with oscillatory vorticity fields, Comm. Pure Appl. Math., 43 (1990), pp.821-853.

[4] W. E AND R. V. KOHN, The initial value problem for measure-valued solutions of a canonical2 2 system with linearly degenerate fields, Comm. Pure Appl. Math., to appear.

[5] W. E AND D. SERPE, Corrector results for the homogenization of scalar conservation lawswith oscillatory forcing terms, Asymptotic Anal., to appear.

[6] B. ENGQUIST, Computation of oscillatory solution to hyperbolic differential equations, 1986,unpublished.

[7] B. ENGQUIST AND T. HOU, Particle method approximation of oscillatory solutions to hyperbolicdifferential equations, SIAM J. Numer. Anal., 26 (1989), pp. 289-319.

[8] D. FORSTER, D. R. NELSON, AND M. J. STEPHEN, Large-distance and long-time propertiesof a randomly stirred fluid, Phys. Rev. A, 16 (1977), pp. 732-749.

[9] J. GLIMM, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. PureAppl. Math., 18 (1965), pp. 697-715.

[10] P. L. LIONS, G. PAPANICOLAOU, AND S. R. S. VARADHAN, Homogenization of Hamilton-Jacobi equations, 1987, unpublished.

[11] T. P. LIU AND Z. P. XIN, Over-compressive shock wave, in Proc. Workshop on Mixed TypeEquations and Phase Transition, Minneapolis, MN, Vol. 27, M. Shearer, ed., Institute forMathematics and its Applications, University of Minnesota, Minneapolis, MN, 1989.

[12] Z. SHE AND B. NICOLAENKO, Temporal intermittency and turbulence production in the Kol-mogorov flow, preprint.