convergence of the random vortex method

Convergence of the Random Vortex Method

JONATHAN GOODMAN Courant Institute

1. Introduction

We prove a convergence theorem for a random vortex method for the incompressible Navier Stokes equations in two space dimensions. Since the literature on vortex methods is extensive we offer only a few remarks here. Although the vortex method predates computers (see [17]), there are two modifications, due to Chorin [6], that are central to this paper. The first is smoothing out of point vortices for numerical stability. All of the convergence proofs for the inviscid Euler equations to date, [l], [4], [5], [8], [lo], rely on smoothing, mainly to insure that certain sums are well approximated by integrals. The second modifica- tion is the addition of random walk to simulate the action of viscosity. We are particularly interested in the case when the viscosity (diffusion rate) is quite small, so we want a proof that allows for randomness, but (unlike the “propagation of chaos’’ theorems; see [13], [14], [16], [21]) does not use the smoothing properties of Brownian motion. Since the Navier Stokes equations are nonlinear, it seems unlikely that the numerical approximations converge weakly if they do not converge strongly. For a random method, this suggests that if the method converges in the mean (the expected value of the computed velocity is close to the true velocity), then the error of any given run is small with high probability. Our theorem is an extension of work by Marchioro and Pulvirenti [13], who used more smoothing than we do and did not discuss some of the above questions, and work of Hald [ll], who proved stronger theorems for some one-dimensional model problems.

Let x = ( x , , x2) E W 2 be a generic point, and u ( x ) = ( u , ( x ) , u 2 ( x ) ) be a velocity field. The incompressible Navier Stokes equations are

u, + ( u v ) u + vp = VV%, v u = 0,

where v is the kinematic viscosity. The vorticity, o(x, t ) , is given by

au au ax, ax, o = curl(u) = v x u = 2 - 2.

The velocity is determined from the vorticity through the Biot-Savart law

Communications on Pure and Applied Mathematics, Vol. XL, 189-220 (1987) 0 1987 John Wiley & Sons, Inc. CCC 0010-3640/87/020189-32$04.00

190 J. GOODMAN

where K , the Biot-Savart kernel, is given by

The vorticity satisfies

(1.3) 0, + (24 v)w = VV%,

and (l.l), (1.2), (1.3) are equivalent to the Navier Stokes equations. We shall often use the Fourier transform

For future reference, we note that

Suppose that w ( x , O ) is a probability density, and that we choose a point, Y(O), with that density:

P(Y(0) E A ) = / w ( x , O ) dx. A

Then (1.3) is the Kolmogorov forwards equation for the stochastic differential equation

(1.5) dY( t ) = u( Y( t ) , t ) dt + 6 dB,

where B ( t ) is normalized Brownian motion; see [2], [7], [20]. That is, Y ( t ) is a random point whose probability density is w ( x , t ) :

P ( Y ( t ) E A ) = / w ( x , t ) dx. A

Now consider R independent random processes, Y,( t ) , . this way. If R is large, then the empirical density function

, Y,( t ) , constructed in

should be an approximation to the true distribution, w ( x , t ) , in the sense that,

CONVERGENCE OF THE RANDOM VORTEX METHOD 191

for any sufficiently regular function q ( x ) ,

In particular, if K were sufficiently regular we could approximate (1.1) by

(1.7) u ( x , t ) K ( x - T ( t ) ) . l R

~ ( x , t ) = \X(x -y)f(y, t ) dy = j = 1

The simplest version of the random vortex method is to replace u by u in (1.5), that is:

(1 4 d Y , ( t ) = u ( Y , ( t ) , t ) dt + h d B , ( t ) ,

where the B,(r) are independent Brownian motions and u is given by (1.7). The T ( t ) are no longer independent, but we can hope that they are asymptotically independent in the limit as R -+ KJ (propagation of chaos).

The method we consider differs from (1.7), (1.8) in two ways. In (1.7) our method replaces the empirical vorticity distribution f by a smoothed distribution f,, and also the continuous time random process (1.8) by a discrete time approximation. We first discuss the smoothing. Let + ( x ) be a Schwarz class function (see [9]) with integral one (a “mollifier”):

with the symmetry

+( - x ) = + ( x ) for all x E I W ~ .

For any E z 0 define the smoothed empirical vorticity distribution, f,, by +Jx) = (1/E2) + ( X / E ) and

Replacing f by f, in (1.6) gives

where

192 J. GOODMAN

This form of smoothing is due to Hald [lo]. Since + is symmetric and K is skew-symmetric, K , is also skew-symmetric and K,(O) = 0. Ths will be used in proving Lemma 2.

To discretize in time, choose a time step 6t > 0 and let t ( " ) = n 6 t . If Z ( t ) satisfies

dZ( t ) = g( Z ) dt + 6 dB,

our approximation, Z(") = Z ( t ( " ) ) , will satisfy

where the N ( " ) are independent normalized Gaussian random numbers; see [7]. Sethian [18] has pointed out that more accuracy is possible using higher-order integration formulas (such as Heun's method, a second order Runge-Kutta formula). This is discussed in Section 2.

The random vortex method is not restricted to the case where w is a probability distribution. Suppose a ( x ) is a probability distribution and u(x) is a bounded function so that

w ( x , o ) = u ( x ) . ( x ) .

Then we may choose the initial positions, Xj(0), of the particles according to the density a but give them vorticity strength

u ( q ( 0 ) ) = Uf.

With all these modifications, the method we study is given by

where the Nj(" ) are independent standard normal random numbers. We prove that with high probability this method produces good approxima-

tions to the true velocity. The proof assumes that the parameters 6t and E are related to R. There are no serious restrictions on at; as long as 6t + 0 as R + co and 6t 2 CR-P for some p > 0 the method will converge. The choice of 6t is thus determined by accuracy and economy considerations. However, we require a large amount of smoothing:

E = R-P where 0 < p < $.

We have not been able to prove (what we believe to be) the correct condition, 0 < p < $. In two dimensions, a regular lattice with grid spacing h has R =

O ( h - * ) grid points. Our requirement corresponds to E z=- hl/' while the correct


condition should be E >> h (the core size is large relative to the distance between particles).

THEOREM. Let u ( x , t ) satisfy the Nauier Stokes equations with initial uorticity distribution w ( x , O ) f S (the Schwartz class). With probability one we have, for any T < 00 and bounded open set A ,

where

and t ( " ) = n S t , provided that E 2 CR-'/410g2(R).

The error in the vortex method has many sources and depends on many parameters. Errors are introduced through viscous splitting, approximate solution of the inviscid Euler equations, smoothing, and sampling. Errors in each stage can be reduced by developing more sophisticated numerical methods, a process that is not nearly finished. Therefore, we feel that it would not be enlightening at this point to combine these diverse errors into a grand error bound. However, some quantitative bounds on sampling error (the main subject of this paper) appear in Section 4. The papers [3], [5], and [l], respectively, have detailed discussions of splitting, smoothing and time discretization errors.

The proof follows the proof given by Hald [lo] for convergence of an inviscid vortex method. First (Section 2), we study the effect of smoothing and time discretization alone. Although smoothing and time discretization introduce possi- bly large errors, they do not induce undesirable effects such as instability or roughening of the vorticity or velocity fields. In Section 3, the consistency part of the proof, we consider independent particles moving with the velocity field constructed in Section 2. Given sufficient smoothing, these independent particles satisfy the equations (1.10) up to an error for which we find crude bounds. It is interesting that the factor 1/R in (1.10) leads to a slightly biased estimate of u( ?(")(t) , 1 ) . An unbiased e$timate results from replacing this by 1 / ( R - 1). The last step (stability-Section 4) is to show that the approximate solutions constructed in Section 3 are close to the exact solution in the I , norm used by Hald:

$(")I 2 5 R'l2log2(R).

This uses a bound for the 1, norm of the linearization of the inviscid part of (1 .lo), which is an exercise in elementary pseudodifferential operator calculus.

194 J. GOODMAN

2. Time Discretization and Smoothing

We need first to study the formal limit R -+ 00 of the vortex method. This separates the systematic truncation error, caused by smoothing and time discretization, from the random errors caused by sampling. This section follows parts of a paper of Beale and Majda [3]. Therefore, we give details only when they differ from those in [3]. Sections 3 and 4 are concerned with bounding sampling errors while this section discusses only truncation errors. This separation, aside from its pedagogical advantage, has a technical motivation. In Section 4 it is harder to justify the linearization when the error JJX - Y/J, , is large; this would be even harder if X - Y included truncation error as well as sampling error. Although the material in this section is quite routine, I must apologize for the many irrelevant technicalities that I have been unable to avoid, and I urge the reader to take the results of this section on faith.

Let us derive the discretized equations. Suppose X E R2, and u E R are random variables with density o(x). That is,

for every bounded open set A . Here x A is the characteristic function of A . Let u(x) be a smooth vector field so that the mapping y = x + St u(x) is a smooth homeomorphism for all S t small enough. If Y = X + St u( X), then Y, u have the density function p ( x ) defined by

(2.1)

To see this, define by

p (x + Stu(x)) = det-'(Z + stDu(x))w(x).

A= { y : y = x + Stu(x) forsomex E A } .

Then X A ( Y ) = x A ( X ) (a tautology), so

Thus, p is the correct density function if

But (2.1) has that property since

& p ( y ) dy = / p (x + Stu(x))det(l+ S t D u ( x ) ) dx = L w ( x ) dx. A

Suppose that we have infinitely many independent YJ(O), uj with density


d 0 ) ( x ) = o ( x , 0). If do) = do) * K,, and Y('12) = Y o J + 6t ' ('1 ( J Y C o ) ) , with infinitely many particles, presumably we get & ) ( x ) (see (2.3)) exactly. In any case, the ?t1l2), uJ are independent and have density y('). Similarly, if y/(') = Y/(l12)

+ mN,('), the qC0) being independent standard normals, then the Y / ( ' ) , a, have density a(') given by (2.4). Continuing in this way, we construct y/("), uJ with density w ( " ) ( x ) = w(x, t '")) , where I(") = n 6 t , satisfying the discretized equations

(2 -2)

(2.3)

u ( " ) ( x ) = K,* o( " ) ( x ) ,

y("+')(x + 6 t u ( " ) ( x ) ) = det-'(l + S t D u ( " ) ( x ) ) d " ) ( x ) ,

(2.4) , ( " + l ) ( X ) = H ( f " t ) p ' " + " ( x ) .

0, = v 2 w . The operator H ( s ) is the solution operator at time s of the heat equation

We shall need to control a(") in weighted Sobolev norms

When p = 0 we have the ordinary Sobolev norm I l f l l , = I \ f l l s , o . The desired estimate is

(2.5) II~("+l)lls,p 4 (1 + C ~ t ) l l ~ ( " ) I l s , p ,

where C depends only on p, s, and ( I ~ ( " ) l l ~ , ~ . From (2.5) it follows that the "semidiscrete" equations (2.2)-(2.4) are stable in that they satisfy

II a(") I IS, p 4 eC'(")II 0 ( 7 0) I I S , p'

where we have used the Sobolev lemma in dimension 2.

196 J. GOODMAN

Starting with (2.2), we note that the bounds IIDu(")lls 5 CIIw(")lls 5 C l l ~ ( " ) l l ~ , ~ follow upon taking Fourier transforms. We also have ~ ~ u ( n ) ~ ~ L m 5 C l l ~ ( " ) l l ~ , ~ since

We break (2.3) into two steps by defining (drop the (n) superscript for now)

+ ( x ) = d e t - l ( I + S f D u ( x ) ) w ( x ) .

If St llDul12 is small enough, then det- '(I+ StDu(x)) = 1 + S t u ( x ) , where llvllS s CIIDulls. Thus, if S r 1 1 0 1 1 ~ is small enough, then

IIJ/lls., 6 (1 + C~t)l141s,,,

where C depends on 1 1 0 1 1 ~ , ~ .

f < x + Stu(x)) = f ( x > for any function f ( x ) . Our first goal is

(2.6)

To study the second half of (2.3), we consider the operation f + f , where

l l f l l o , p 6 (1 + C~f)llfllo,,,

where C depends on IIuIIL, and IIDuIIL,. We have

= / f 2 ( x + S t u ( x ) ) ( l + I x + S t u ( x ) I2)'det(l + S t D u ( x ) ) dx.

But Idet( I + S t Du( x))l 5 1 + C S t , and, by the mean value theorem,

(1 + ~x + s t u ( x ) 12)'s (1 + cst)(l + /x12)',

which proves (2.6). We now verify by induction on k that

(2.7) Ilf l lk, ' =< (1 + cwllfllk,, + Cat lIfll2,'llDullk.


The cases k = 1 and k = 2 are like the case k = 0, so we assume that k > 2. Now ?,j= qf* (I + 8 t h ) -l, and (I + 6tDu)-' = I + 8 t u , where /lullk CJJDulJ, if IJDuJJ, is not too large. Thus,

By the induction hypothesis,

To use interpolation inequalities (see e.g. [19]), set 3 = X ( k + 1) + (1 - A) 2 and k = p (k + 1) + (1 - p ) 2. Then

and

A calculation shows that X + p = 1, so, for positive numbers a and b, axbp C( a + b). Another calculation then yields

An interpolation calculation of this kind also leads to

Altogether, this verifies that

where C depends on IIDullz. This completes the inductive proof of (2.7). Finally, since p = 4, (2.7) specializes to the desired bound,

llPlls,p 5 (1 + C~~)l141s,p.

Finally, step (2.4) requires weighted norm bounds for the linear heat equation, a, = V'O. Multiplying by (1 + I X ~ ~ ) ~ W , integrating over x, integrating by

198 J. GOODMAN

parts twice, we get

The same estimate holds for dfw for any a, so we have

which completes the proof of (2.5). There are two remaining points, which have the same resolution, accuracy of

this " semi-discretization", and verification of the low norm bounds used above. One can check (see below) that w ( x , t ) , the solution of the Navier Stokes equation, is bounded for all time in the weighted norms, once w is known to be bounded in unweighted norms. Write (2.2) and (2.3) in the symbolic form

p("+') = Ad(6t, ~ ( " ) ) ~ ( " ) ,

where Ad(&, w ) is the advection operator for time 6t using the velocity field u = K , * w. In this notation, (2.2)-(2.4) become

A Taylor series calculation gives

where

for p 2 2. Thus, we have nearly proven:

PROPOSITION 1. I/ 1 I ~ ( * , O ) l l ~ , ~ < ccfor ( s , p ) large enough ((s, p ) > (5,2)), then, /or all T < cc there is a C ( T ) such that

Proof: Given the above, the proof is carried out as in [3]. We just need to verify that ]la(*, t ) ] l s , p is bounded, starting from the hypothesis (a known (see e.g. [3]) theorem) that l l w ( * , t) l l , and [la'%(*, t)llL,, 0 5 5 s - 1, arebounded.


We have the estimate (using v u = 0),

+/V'(l + IxI">" o2 - 2/(1 + ~x~2)pv2zw

Starting from the (0,O) bound, this gives a (0, p ) bound for any positive integer p . Furthermore, taking 8: with la1 = s of the Navier Stokes equation and estimating as before gives

d 2 2 , I Ia34*J) i lo,p 5 c c l l ~ : ~ ( * , ~ ) l l L , l l 4 ~ , t ) l l s , p I c r l j s - 1

which implies boundedness of 11o(*, r ) ) l s , p , and completes the proof of the proposition.

For later purposes, we discuss the probability densities, T(")(x), of a single particle moving in the velocity fields u ( " ) ( x ) constructed above. The T(") are defined by

E[XA(Y'"')] = /n(")(x) d x , A

where the Y(') E R 2 satisfy

when t ( " ) 5 T. We conclude with two remarks on accuracy. First, Beale and Maida [ 5 ] show

how to choose mollifiers, cpk, so that E is replaced in (2.9) by ek for any k . Second, if we replace (2.8) by the Strang splitting

200 J . GOODMAN

we get lip(*, t (")) l l 5 C ( 6 t 2 + E ) , provided that is a second-order accurate approximate solution of the inviscid Euler equations. This reinforces Sethian's suggestion to use Heun's method in place of forwards Euler in the advection step.

3. Construction of Approximate Solutions

The consistency step of the proof is to use the velocity fields u ( " ) ( x ) defined in Section 2 to construct approximate solutions of the discrete vortex equations. The vortex method equations (1.10) have the general form

The approximate solutions, Ycfl), are defined by YcO) = and

where the d") are the velocity fields constructed in Section 2. Note that ?(") is independent of Yk(") if j # k.

The main purpose of this section is to show that the Y ( " ) satisfy

where the residual, p'"), is small in the I, norm,

From Section 2 we have the following situation. For each n, the q(") are independent samples of the probability distribution T(")( x), and, for any bounded continuous function a(x) we have

E[ u ( y ( " ) ) o J ] = /u (x )d ' ) (x ) dx.

Thus pj") is the average of many independent identically distributed (i.i.d.) random variables, each with expected value zero.

Several simple points of probability theory are important here. Let BJ be bad sets with small probabilities P ( B J ) 5 aJ, and let B = U J B J be the event that one of these bad events occurred. Then P(B) 5 EaJ. The lemmas below bound the


probability of certain bad events (bad approximation of sums by integrals) by CR-P. Since there are finitely many such lemmas, all the inequalities in all the lemmas hold except on a bad set of probability at most CR-P. The second point concerns conditional probability. Let Z be a random variable with probability measure d p ( z ) , and B a bad set. If P ( B I 2) 5 C independent of Z , then (see [7]) P ( B ) = l P ( B 1 z ) dp(z) 5 C. The final point is the Borel-Cantelli lemma; see [7]. The probability that the error violates the bounds given here and in Section 4 is at most CR-p. Therefore, if p > 1 the probability that the bounds are violated for infinitely many R is zero. This justifies the almost sure convergence stated in the theorem.

Actually, there are several sums over the random points Y,("), such as (3.2), that need to be approximated by integrals, All these estimates are consequences of

LEMMA 1. Let Z,; . ., Z , be i . i .d. random variables with E ( Z j ) = 0,

E ( Z f ) s g ( R ) , g ( R ) -+ m as R + m,

and lZjl 5 C \ / M . Then the sample mean 2 = (1/R)C7-,Zj satisJes

(3.5)

and

where Cp is a constant depending on C and p > 0.

Proof: Since the Zj are i.i.d., we have

e = .[..P( / 5 2 ) ]

From the Taylor series expansion

e s = 1 + s + $s2ea ,

202

with (ul =< 131, we get

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which implies that Q 2 (1 + C / R ) R 5 C. Note that this can be repeated with each Zj replaced by - Z j , so (3.5) follows from the inequality elSl 5 es + e-’. Now, if + is a positive random variable with mean E(+) = p < M), then Markov’s inequality is

Upon taking m = RP in ( 3 . 9 , we get (3.6). This proves Lemma 1.

We are now ready to bound

where ZJ = u ( Y l ) - K,(Yl - T)uJ, and we have used K,(O) = 0. Recall that (Y,, uJ) is independent of (Yk, uk) if j f k. Consider (Y,, a,) as given and let E‘[ -1 denote the conditional expectation E‘[*] = E[*)(Yl, ul)]. In Section 2 we showed that

E’[K,(Yl - ? ) u j ] = JK,(Y, - X ) U ( X ) dx = u ( Y , ) .

To use the method of Lemma 1, we need a bound for the variance

E‘ [ (Z j ( z ] = E ’ [ l u ( Y , ) - K,(Y, - q)uj12] .

But u is bounded, u is bounded, and Y, could be any point, so (without loss of generality) it suffices to bound

With the information from Section 2 (smoothness and decay of T), this follows from

CONVERGENCE OF THE RANDOM VORTEX METHOD

Using the Fourier transform

203

where &([) = $ ( E [ ) , the simplest bound is

For larger x values, we integrate by parts: *

With these bounds (always assuming that E 5 i),

as desired. Under the modest assumption that E 2 R-'/', the hypotheses of Lemma 1 are satisfied with g( R ) = 8 log3( R); the conclusion being a bound on pl:

The same bound must also apply to the other p j , j = 2; . -, R. If we further assume that 6t 2 CR and note that there are R points in all, we get

*See note added in proof.

204 J. GOODMAN

LEMMA 2. t ( “ ) 5 T , then

There is a constant Cp depending on p , the flow, and T, so that if

for some n 5 N , j 5 CR-(p-2), 1 CPlog5l2( R)

and,

for some n 5 N CR-(p-2). 1 CJog ’ j 2 ( R ) fi

In the stability proof of Section 4 we shall need estimates for three more sums of this kind. We make these estimates here, but the reader may want to skip them until they are used in Section 4. The first sum is

/

where again the superscript ( n ) has been suppressed. As above, the main point is to bound (still setting j = 1 without loss of generality)

Since I $ I ~ ( X ) ~ 5 1 / ~ ~ , Lemma 1 will give a useful bound on sj if E is large enough, namely

E 2 CR-1/210g2( R ) .

In this case, we can take

then Lemma 1 implies that

Taking unions of bad sets, we see that, except for a set of probability at most


CR-(P-2), we have, for all j and t ( " ) 5 T,

205

The conclusion that we shall need is only:

LEMMA 3. The bound

(3.7)

holds for all j and t (") 5 T, except on a set of runs with probability at most R-(p-2) .

Our last two sums are related to the linearization of \k about the approximate solution, Y. Let L,(x) = d ,K , (x ) be the 2 X 2 matrix with Fourier transform (in view of (1.4))

In the linearization, we shall need to consider sums evaluated at general points, not just at one of the 5. For this purpose, define r ( " ) ( x ) by

(3.8) r ( " ) ( x ) = 1 "

L E ( x - y/("))o, - / L , ( x - y ) w ( " ) ( y ) dy. j= 1

The distinction between division by R and division by R - 1 is immaterial since we are only interested in rough bounds on

j = l

To bound S, we need only bound

since the integral contribution to r ( x ) is bounded if w ( x ) is regular enough. We shall use the fact that

206

where

L , ( x - I y ) U j j = 1

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6 c,

If

then Lemma 1 implies that

E[exp( I f i P ( 6 ) I ) ] 5 c* To apply Jensen's inequality (see [7]) set

so that

and, since z + e x is a convex function,

and finally, using (3.9),

If we use Markov's inequality as in Lemma 1 with m = RP, and assume that E 2 R-1/410g( R) , then there follows

P(s 2 C,log-'(R)) 5 R-P.

This concludes the proof of:

LEMMA 4. Except for a set of probability at most R-(*-'), we have

where C depends on theflow and on t (" ) 5 T.


Let M , ( x ) be one of the entries in L,(x) , and let A be the R x R matrix with entries

1 a . J k = - - -M(Y, - R - 2 Yk).

In Section 4 we shall need to evaluate

These sums are approximated by integrals, so we bound the error

the terms omitted from the sum being irrelevant. The goal will be

LEMMA 5 . Except on a set of probability at most R-(pP3) we have

k = l

if E 2 R-'/410g2(r). Here C is uniform in t ( " ) 5 T

Proof: As in the proof of Lemma 2, we apply Lemma 1 to a conditional probability distribution. Let E'(*) = E(*l q, Yk) be the conditional expectation given y and Yk, and let

Given y/ and Yk, the Z, are i.i.d. if 1 # j and I f k. Thus, we need to estimate

g (R ; y , r') = E( z: 15 = y , y, = Y ? )

5 / M : ( x - y )M, ' (x - y ' ) . ( x ) dx + C

I - C J M ? ( x - y ) M ? ( x - y ' ) dx + C

= C h ( y , y ' ) + C .

208 J. GOODMAN

Now, h ( y , y ’ ) only depends on Iy - y’l, so we take y = ($d,O), y’ = (- f d , O ) with d = Iy - y’l. Then

h = 2J M,z(x - y ) M , z ( x - y ’ ) dx X l b O

First we have

We got two bounds for IK,(x)l by integration by parts no times or one time. Here, the analogous bounds for IM,(x)l come from integration by parts zero or two times. They are IM,(x)l b,(lxl), where

With these bounds we may take

If E 2 R-‘/210g2(R), we have lZ,l 5 C / m and Lemma 1 yields

P[ l q k l 2 Cb,( I ? - Y,l)l~g-~( R)] 5 RPP.

Again taking the union of O(R3) bad sets, we get, except on a set of probability at most R-(p-’)

for all j , k , and t ( “ ) 5 T. Lemma 5 now follows from bounds for


The necessary bounds are as follows

I = / b , ( l Y , - x l ) n ( x ) d x j C + C b,( lx l )dx i x , , 1

In a similar way, the variance satisfies

c log2(1/&) /b:( IYJ - x l ) n ( x ) dx 5

E 2

Thus, if E 2 R-'/210g2(R), we have IS,^ 5 C log-'(R) with high probability. Finally,

5 c(iog2(R) + c ) ~ o ~ - ~ ( R )

I - c log-'@).

Since (l/R)Cj(rjkl has similar bounds, this proves Lemma 5.

4. Stability and Convergence

In this section, we give an error bound for the computed velocity field. The set of runs for which this bound does not hold has small probability. The situation is as follows. The exact solution to the discrete equations, X ( " ) , satisfies

X("+l) = X ( " ) + 6t \k( XC")) + r n N ( " ) ,

Y("+1) = Y(") + 6t \k(Y'")) + r n N ( " ) + 6tp'"' .

and the approximate solution analyzed in the previous section satisfies

Moreover,

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with high probability. From this, we want to conclude that *

again, with high probability. It does not follow from (4.1) that, say, particle X i " ) is close to its counterpart Yifl) , but the majority of the particles must be close. Note that if (4.1) holds and R is large enough, then

The bounds (4.1) and (4.2) are proven by induction on n. Assume that they hold for n and subtract the Y equation from the X equation:

X',+l) - Y("f1) = X'" , - y(") + & ( q r ( X ' " ' ) - \k(Y '" ' ) ) - a t p ( " ) .

Our object is to show that if (4.2) holds, then

(4.3) I( \k( X ' " ) ) - \k( Y @ ) ) I[,, 5 CII X'" ) - Y(") 11/,,

where C depends only on the flow and on t , 5 T. The estimate (4.1) follows from (4.3) in a standard way.

We begin with several preliminary reductions that justify linearizing about the approximate solution, Y(") , and neglecting one of the terms in the linearization. First set

f, = \kJ ( x@)) - \k, ( Y ( " ) )



To have 1r-l 5 C, we tentatively assume that

(4 -4) E 2 R-'I6log(R),

but given this, we shall see that

Here, li?,L,I is a generic upper bound for second partials of K,. A typical such partial is NE(x) with

where p 3 ( 5 ) is a homogeneous polynomial of degree 3. As in Section 3, integration by parts zero or three times gives the bounds

(4.6)

Lemma 4 bounds the first term on the right in (4.9, while the Cauchy-Schwartz inequality, (4.2), (4.4), and (4.6) bound the second term:

This clearly gives

51 5 c + c l o g 4 ( R ) 5 c.

as desired, and eliminates the gJ from further consideration. The second reduction involves h:

212 .I. GOODMAN

Given (4.2) and (4.4), the h Z j are easy to estimate if IhZj( 5 ( C / E ~ ) ~ ~ X - YI& so that

c llh211,z 5 ?IlX - yll,211X - Ylh, 5 CIIX

E

E 2 R-'/610g(R) since

- y11,2.

Thus, (noting that lujl C ) proving the stability bound (4.3) reduces to proving

LEMMA 6 . With high probability there is a C so that if

then l141,z 5 Cllall,2-

Proof: We must show that the matrix whose entries are the 2 x 2 blocks L,(y - Yk) is bounded in 1,. By bounded we mean bounded with a constant independent of R with high probability. This is equivalent to the boundedness of the matrices with scalar entries Me( q - Yk), where M , ( x ) is one of the entries of L,(x) . Let A be one of these matrices: A = {a , , } , where a,, = M e ( q - Yk). In order to show that A is bounded, we shall find matrices B and C such that

(4.7) D = A'A + B'B + C'C

is bounded. This implies the boundedness of A since B'B and C'C are non-nega- tive. The matrices B and C are chosen so that D is a close approximation to the identity, close enough so that its boundedness is clear. This argument is moti- vated by Hormander's proof of boundedness of pseudodifferential operators in

We shall often approximate matrices by others. In t h s process, we shall use Young's inequality (see [9]) to check that the error is a bounded matrix. If F = G + H are R X R matrices, then F is bounded if G is bounded and H = { h j k } satisfies, for all j , k ,

P21.

C l h j k l 5 c, C l h , k l 5 k j

(4.8)

We will be able to neglect errors in approximations if the errors satisfy Young's criterion, (4.8). Our first application is to approximate A'A by F, where

Lemma 5 states that the error in this approximation satisfies (4.8) and therefore is negligible.


The main step is to show that F = G + H, where H satisfies (4.8),

213

and

This will be an application of elementary pseudodifferential operator calculus (see [21] or [22]). Granted this, the proof of Lemma 6 is immediate. Depending on which entry of L, we took for ME, q ( x ) will have Fourier transform (in view of the expression for 2, in Section 3)

or

If we take ME to be the ( i , j ) entry of L,, then g,(x, y ) = s ( x ) q Z j ( x - y) . Thus, if A comes from the (2'1) entry, B from twice the (1,l) entry, and C from the (1,2) entry, then in (4.7) (except for negligible errors) D = { d j k } , where

The symmetry condition +(-x) = + ( x ) implies that 6 is real, so that i ( 5 ) = ( 1 / 4 ~ ~ ) & ~ [ ) ~ , and

If @ is of Schwartz class, then so is r , and the proof of Lemma 3 applies to r as well as $I. In fact, the proof did not use smoothness, only the variance estimate (which also applies to Ir(x)l). But this is exactly what is needed to show that 11011,, is bounded by Young's criterion (4.8). Note that (3.7) still holds if we sum over j instead of k.

214 J. GOODMAN

It remains only to study H = {h,,}, h,, = h ( Y , , Y,), where h(x, y ) = f ( x , y ) - g(x, y ) (see (4.9) and (4.10)) and to show that H is bounded by (4.8). To use the technique of the proofs of Lemmas 2 and 3, we need constants depending only on the flow so that *

(4.13)

Once (4.11)-(4.13) are verified, both bounds on (4.8) follow since the argument is symmetric in x and y .

We compute a partial Fourier transform

= j M E ( z - x)T(z)e- iq*xkE(-q) dz

= 4 n 2 e - ' ~ 4 ( ~ , + I , + z3)kE(-q),

where, for k 2 1,

* S e e note added in proof.


When 151 2 31171 we have h?e(q - 5 ) = &(q) + m(q, 5 ) , where

21 5

s c1711-'11~l13. Finally, for

we have I3 = m(x)de(q) + O ( ~ T J ~ ' - ~ ~ ~ T ~ ~ ~ ) . This yields the desired conclusion

To get (4.11) from this, use the Plancharel identity: *

A calculation verifies (4.12):

For (4.13), note that I1,l 5 CJq)-31)nJ14, so that, after integration by parts, we get

as claimed.


216 J. GOODMAN

We now sketch a more careful argument that only requires E 2 R-1/410g2( R). Start by dividing the particles into bad ones, where Xj is far from 5, and good ones, where Xj and 7 are close:

G = [l;.., R] - B .

Given (4.2), B cannot be too large (this is Chebyshev's inequality):

IBI 5 R'I21og4( R).

Consider the second term on the right of (4.5). Let N, be one of the partials in axLe. We fix x and try to bound

By the intermediate value theorem,

lp j - (y, - x) I 5 Iy, - 31.

In view of (4.2) the necessary bound is

(4.14)

The contribution of B to S is not too large (recall (4.6)):

For v = 1;. ., R , let

G, = { j E G : Ipj - XI 5 V E } .


Then, using both parts of (4.6), and Abel summation by parts, we have

But IGvl can be bounded using a combination of Lemmas 2 and 3. Let a ( x ) be a positive Schwartz class function with a ( x ) >= 1 when 1x1 1, and set a , ( x ) = ( l / ( v ~ ) ~ ) a ( x / ~ v ) . Set

Since Gv is contained in G and E 2 R-'l4, IGvl 6 IHv+II. But

for all x and v E (1; ., R ] with high probability. Thus IG,l 2 IHv+lI 6 C E ~ V ~ R . Substituting into the above yields the desired bound.

in the second preliminary reduction above. Suppose

A similar bound holds for the h

Then

Setting S = maxk(l/R)z,N:(p,k), the required estimate is S I I X - ~ 1 1 : 2 - I C, or using (4.2), S CR-'log2(R), which we just verified above.

We conclude by giving some error bounds for the computed velocity field. A bound for the weak error is easy and sharp. By contrast, our bound for the

218 J. GOODMAN

pointwise error is harder and less sharp. The pointwise error, e ( x ) , is given by

For the weak error, let f ( x ) be a Schwartz class function and consider

From the proof of Lemma 4, it follows that I ( f , e,>l 5 CR-'/210g5/2(R). Thus, we need only bound (f, e , ) , where

Now,

But, (1.4) shows that Ik,(5)1 5 C/(51, and l e iEex - e iEaY I = < 151 ' Ix - Yl? so

For the pointwise error, let Q be a square with side A . We shall show that, with high probability,

(4.15) sup I e ( x ) I 5 C(Q)R-'/410g(R). X C Q

We may assume that Q is aligned with the coordinate axes and centered at the origin. Set ( x j , y k ) = ( j , k ) R - * and let QR = ( ( x j , y k ) E Q ) . The first step is to show that (4.15) holds with Q replaced by QR. This follows the argument used to bound S in (4.14). We have IK,(x - X, ) - K , ( x - ?)I 5 JL,(pj)l 13. - ?I; thus, if we can show that ( M , is as in Section 3)

(4.16) l R CR1l2

log2( R) ' w = C M 3 P j ) 5 ;=l

we will have


Using (3.10), we find that

C M:( pi) C ~ - ~ R l / ~ l o g ~ ( R ) 5 CR3/210g-4(R), j € E

and

I - CR l ~ g - ~ ( R ) l o g ~ ( R ) R - ~ / ~ l o g ~ ( R ) R + CR

I - c l o g 2 ( R ) R ~ / ~ .

This proves (4.16) and completes the proof of (4.15) (since lQRl 5 CR24 and p may be arbitrarily large).

To complete the proof, note that

But clearly Ive (x ) l 6 CR-P for some p > 0, so it suffices to take q 2 p.

Acknowledgment. I am indebted to Henry McKean who told me that “sharp estimates come from estimating exponentials”. This advice vastly simplified Section 3.

The research for this paper was supported by the National Science Founda- tion under NSF Grant No. DMS-8501953.

Note added in proof: K. Meth pointed out that many of the factors of log(l/E) are unnecessary. This is true in the bound for lK , (x ) ( on p. 203 and in (4.13). With this, (4.11) follows directly from (4.12) and the corrected (4.13), so the argument on p. 215 is unnecessary. This also improves log5l2(R) to log3l2(R) in (4.1). To remove the log(l/&), we integrate by parts an additional time and use the fact that I V $ ( E ~ ) ~ = 0(~~151), which holds since 4 is real and symmetric.

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convergence of the random vortex method

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