new stability estimates for the 2-d vortex method

New Stability Estimates for the 2-D Vortex Method

JONATHAN GOODMAN AND THOMAS Y. HOU Courant Institute

Dedicated to Natascha Brunswick with whom it has been a pleasure to work.

1. Introduction

In this paper, we simplify and extend the convergence theory for the vortex method for the two-dimensional incompressible Euler equations. All of the previous theory relies on the fact that V K (the derivatives of the Biot-Savart kernel) is a kernel of Calderon-Zygmund type (see, e.g., the review papers [4] and [l 11). Here, we use only the mapping properties of K itself. As a result, we can prove stability in the maximum norm. Actually stability in the maximum norm would allow us to analyze convergence of the vortex method with local regridding. This will be studied in a later paper. We can also prove convergence for systems where K is replaced by more singular kernels, in particular the case where K itself is made of Riesz transforms or other kernels of Calderon-Zygmund strength. A similar convergence result was obtained earlier by Chacon and Hou (see [ 5 ]) for a Lagrangian finite element method which applies to the 2-D Euler equations with any L,', kernel K . Although we do not know of applications of these 2-D Euler equations with a more singular kernel, they are well posed (see Appendix) and seem interesting mathematically.

The other aspect of the present argument is that our argument allows for more general initial distributions of the vortices. This is in contrast with previous convergence proofs which typically assume that the initial distribution of vorticity lies on the nodal locations of a uniform grid. In this paper, rather than using the specific locations of the points X I , . * ., X,, we assume that W L ( X ) = Cj 6(x - Xj)wj is a good approximation in some weak sense. We will present two convergence proofs. The first one provides a stability estimate in a maximum norm and gives a sharper estimate on the truncation error in the case of a regular initial grid. The second approach proves convergence in a negative norm. The purpose of the second proof is to avoid using derivatives of the flow map (the Lagrangian coordinate) in favor of derivatives of Eulerian quantities such as vorticity. For example, in the rotating vortex patch, which has been used as a test problem for vortex methods, the derivatives of vorticity do not grow in time while the flow map rapidly becomes very stretched.

The analysis is simple and self-contained. The main idea, which is due to Cottet et al. (see [ 7]) , is to regard the approximate vorticity distribution, a h , as a distribution solution of the Euler equations and derive an equation for the consistency and stability errors respectively. With some new observations, the analysis in [ 71 can be simplified and improved to obtain our desired convergence result.

Communications on Pure and Applied Mathematics, Vol. XLIV, 1015-1031 (1991) 0 1991 John Wiley & Sons, Inc. CCC 0010-3640/91/8-91015-I 7$04.00

1016 J. GOODMAN AND T. Y. HOU

Consider the incompressible 2-D Euler equations in the vorticity stream function formulation (see [ 11 ):

(1.1) wt + (U*V)O = 0 , w(x, 0 ) = w o ( x ) ,

In this paper, we shall study the analogue of the 2-D incompressible Euler equations ( 1.1 ) and ( 1.2) with a more singular kernel K satisfying the following properties:

(i) K ( X x ) = X-( l+' )K(x) for X > 0 and K(-x) = - K ( x ) , (ii) K ( x ) is smooth if x # 0,

(iii) V . K = 0.

For example, K( x ) = ( 1 / 2x I x 1 2 + ')( -x2 , x1 ) is a kernel of this type. Note that for r < 1, K is L,', . But for r = 1, K is a Calderon-Zygmund kernel (see [ 121 ), and each component of K is a Riesz transform. Moreover, property (iii) implies that the velocity field is incompressible. We call the Euler equations ( 1.1 )-( 1.2) with this more singular kernel K the Riesz-Euler equations. We shall always assume that w has compact support.

The vortex blob method consists of representing vorticity as a sum of D i m delta functions, centered at particle positions X,( t ) , which are advected by the smoothed version of the induced velocity from other particles. This was introduced by Chorin in [ 61. It can also be formulated as follows (see [ 71):

where w ! ( x ) = 2, 6 ( x - aj )wjh2, a, = ( j ,h , j2h) , j = ( j l , j 2 ) E Z2, w, = w o ( a j ) and K6 is a regularized kernel obtained by convolving K with a smooth mollifier h.

Outline of Convergence Proof

Following Cottet et al. (see [ 7]), we write w - wh = Ah + ph, where

STABILITY ESTIMATES-2-D VORTEX METHOD 1017

and

The term K6 * ph, which contributes to the stability of the vortex method, can be estimated easily. Define a continuous flow map Xh corresponding to the approximate velocity field u h by

(1.9)

Since uh is incompressible, Xh has an inverse, which we denote by X h l . It follows from ( 1.7) that p h has a smooth solution given by

Thus the smoothness of w and the incompressibility of X h l imply that

(1.1 1)

where the L p norm is defined over a bounded domain since Vw is of compact support.

Note that

if r = 1 by the Calderon-Zygmund inequality (see 1121) and

(1.13) IIK6*p~llLp5 C l ~ ~ & ~ ~ L ~ - ~ ~ ~ ~ ~ ~ ~ ~ ~ C l l ~ h l l ~ ~ , for 1 S P S a,

if 0 5 r < 1 by Young's inequality; see [ 81. Here & is the cut-off K defined by &(x) = &(x) if 1x1 I 2 R and &(x) = 0 if 1x1 2 2 R , R is the radius of the support for vorticity. The fact that p h has compact support and the L p norm is defined over a bounded domain allows us to replace K6 by an L integrable k6 in ( 1.13). Hence we obtain the desired stability estimate

1018 J. GOODMAN AND T. Y . HOU

where 1 I p 5 co if r < 1 and 1 < p < co if r = 1. Note that we did not assume a regular initial grid in obtaining stability estimate ( 1.14).

The term Ks * Ah corresponds to discretization error. It is an easy matter to check that ( 1.6) has a distribution solution given by

Consequently,

To prove convergence of the vortex method, it is sufficient to assume that the initial vortices aj and their strengths wj are distributed in such a way that the right side of ( 1.16) is small, say O(h') for some p > 0. But in the case when the initial vortices lie on the nodes of a uniform grid, the right side of ( 1.16) corresponds to the error from the trapezoidal rule approximation to the approximate velocity integral. Under the assumption that vuh and 6k-2Vk~h are bounded for 2 5 k S 1, which can be justified later, we can show that

Finally, it is well known that for an m-th order cut-off function & (i.e., 4 has integral 1 and m - 1 order vanishing moments) (see [ 3]),

(1.18) IK*w - K ~ * o / I C6",

since w is smooth. Then it follows from (1.8), (1.14), (1.17), and (1.18) that

Hence Gronwall's inequality implies convergence of the method

Additional Remarks

A simpler proof which estimates the vorticity error in a negative norm is also possible with sufficient smoothing. This will be given in Section 3. On the other hand, it would be interesting to see whether smoothing is unnecessary for convergence in the case of an irregular initial grid as it is for a uniform grid; see [ 9 1. Of course, without smoothing close vortices can have arbitrarily high speed, but even


in that case the vorticity distribution is strongly continuous in time, in a weak spatial norm. Consider the point vortex approximation to the 2-D Euler equations,

(1.21)

where K is the Biot-Savart kernel and the X,(O) are distributed arbitrarily. For a smooth test function $, define

Differentiating S( t ) and using ( 1.2 1 ), we get

dS - - - c c V $ ( X , ) K ( X - X , ) W / dt i = I J + r

(1.23)

Interchanging i with j and using K ( Xi - Xi) = - K ( X, - Xi ) gives

Since I K( x - y ) I = ( 1 / 2a I x - y I ) and I V4( x) - V$(y) 1 5 (SUP I V2$ I ) I x - Y I, we get

(1.26)

This might seem surprising since there is no bound for the contribution, dS, / dt = (d/dt)$(X;(t))a,, of a single particle. The inequality ( 1.26) expresses some can- cellation among these terms. No bound like ( 1.26) seems to hold for more singular kernels.

2. Convergence of the Vortex Method in a Maximum Norm

We assume that the mollifier function $ satisfies the following three properties: s $( x) dx = 1, s xa$( x) dx = 0 for 1 5 I a I 5 rn - 1, and $ is radially symmetric.

1020 J. GOODMAN AND T. Y. HOU

Then 46 is defined by & ( x ) = 4(x/6) /S2. The main result in this section is the following convergence theorem for the vortex method.

THEOREM 1. Assume that 4, oo E Cb(R2) for 1 2 2 and 6 = hq for some 0 < q < 1 / ( 1 + r ) . Then the solution of the vortex method ( 1.7) and ( 1.8) for the Riesz- Euler equations satisfies

In the case of 1 = 2 and r = 0, the term ( h/6)'62-' is replaced by h210g( 116). Here u(x , t ) is the exact velocity of the Riesz-Euler equations and 1 < p < co . Moreover, in the case of K E Li, (i.e., r < 1 ), the vortex method is stable in the maximum norm (i.e., p = co).

Remark. The oddness of K (a fact first used by Beale in [ 2 ] ) allows us to obtain one order higher accuracy than the usual accuracy obtained by Cottet et A. in [ 71. Further, our result also applies to initial data which are only Holder continuous (i.e., 0 < 1 < 1 ) in the case of 2-D Euler equations ( r = 0). More precisely, the error in velocity is bounded by C( 6 I + ' + h'), provided that 0 < q < 1. This recovers the result obtained earlier by Hald; see [ l o ] . In this case, the stability estimate ( 1.1 1 ) has to be modified as follows:

Then the Calderon-Zygmund inequality implies (see 71):

The proof of consistency is even easier than the one we present below, since we do not need to desingularize the kernel. It is enough to consider g( a) = K a ( x - Xh( a, t ) ) w o ( a ) and define T* = sup{ t 5 T : IIVuhllLrn S M } in the proof below.

Proof of Theorem 1: We have basically proved Theorem 1 in Section 1 (see ( 1.9)-( 1.21 )), except for estimating the discretization error

which corresponds to the error from the trapezoidal rule approximation to the approximate velocity integral. Define T* as follows


where Mis a constant depending only on the regularity of u. Differentiating ( 1.12) and using the assumption (2.2), we can show that

Since VUh and V X , are bounded for t 5 T*, VK8*Xh is also bounded. Therefore, it is sufficient to bound (2.1) in the case when x = &(aj, t ) . Without loss of generality, we may assume that x = Xh(0, t ) . Since K is an odd function and the cut-off function is radially symmetric, it is easy to see that K6 is also an odd function. Thus we can express &* Ah as follows

(2.4)

where the integration and the summation are over a bounded symmetric domain and

where f is a symmetric smooth cut-off function satisfyingf( a ) = 1 if I a I S R and f ( a ) = 0 for I a I 2 2R, R is the radius of the domain of integration in (2.4). Equation (2.4) is valid because the last term in (2.5) is an odd function of a, and consequently does not contribute to (2.4). It has been shown (see, e.g., [ 1 1 1 ) that the trapezoidal rule approximation gives high order accuracy:

where p > 211, q = p / ( p - l ) , and Bj is a square box with length h , centered at a,.

After some straightforward calculations, one can show in the case of I = 2 that

(2.7)

where C depends on Vwo and the bound of 1 VXh I which is bounded for t 5 T*. It follows from (2.6 ) that

1022 J . GOODMAN AND T. Y. HOU

In the case of I = 2 and Y = 0, the last term in (2.8) is replaced by Ch210g( 1 /a). Arguing inductively, which basically amounts to estimating the I - 2 derivatives of V K 6 ( a ) V 2 X h , we can show that

This proves ( 1.20) for t 5 T*:

Thus the Gronwall inequality implies that

To complete the proof, we need to estimate 1 VkUh 1 for 1 5 k 5 1. Since I Vku I is bounded by some constant C( u ) for t i T and k 4 i, it is sufficient to estimate

1 Vk( u - MI]) I . To this end, we write V k ( u - ul1) as

Since Vku is smooth, we still have

(2.12) ) I ) S C 6 " .

Arguing in the same way as in the proof of (2.9), we can show that

(2.13) I IZZl 5 C(h/6)'62-k-r.

To estimate IZ, we note that

where we have used the fact that llVk K6 )I I 5 C6 - k . This can be obtained easily by using the decay of K (see, e.g., [ 31). In the case of k = 1 and r = 0, we have also used the fact that & has compact support and p = a. Thus L' norm of VK, is


defined over a bounded domain. In any other cases, V k K , is L 1 integrable. Then it follows from (1.14), (2.11), and (2.14) that

(2.15) I ( Z Z ) ) L p S C(Sm + (h /6 )162- r ) /6k+r- , f o r t 5 T * ,

In summary, we have shown that

(2.16) I(Vk(u - uh)llLp 5 C(Sm + (h/6)‘62-r)/6k , for t 5 T* .

Using the Sobolev interpolation formula (see [ 8 I ) , we get for p > 2

Hence we obtain

1 c 6k-21)Vk(u - Uh)JIL” k = 2

(2.18) I C(Sm + (h /6 ) ’62 - r ) /6 (2+2/p ) , for t I T* ,

where p can be arbitrarily large. For 6 = h4 with 0 < q < I / ( I + r ) , the right-hand side of (2.18) can be made arbitrarily small. Therefore, we conclude that

(2.19)

where C( u ) is a constant depending only on the regularity of u. Similarly, we can show that 1) Vuh )I L m 5 M . This implies that T* = T and (2.1 1 ) is valid for the entire interval t 5 T . This completes the proof of Theorem 1.

3. Convergence of the Vortex Method in a Negative Norm

In this section, we analyze convergence of the vortex method with more general initial vorticity distribution. Specifically, we study initial vorticity approximation of the form w i = C,”= F(x - X,(O))w, with bounded initial particle positions and bounded total vorticity (i.e., 2,”- I w, I 5 C ) . The main result in this section is the following convergence theorem in a negative norm.

THEOREM 2. Assume that 4, wo E Cb(R2), l(wo - W ~ ) I - ~ S Chs for some ,6 > 0. Further we assume 6 = h4 for 0 < q < ,6/(4 + 2s) and s 2 2. Then the solution of the vortex method ( 1.7) and ( I .8) for the Riesz-Euler equations satisfies

1024 J . GOODMAN AND T. Y. HOU

for 0 S t S T, where C( T) does not depend on derivatives of theflow map.

Proof of Theorem 2: Subtracting wI + u - Vw = 0 from ( O h ) [ + uh - VWh = 0, we get

It is enough to consider the case where u is replaced by the regularized one K,*w. It can be shown easily that 1 w - w, 1 is bounded by O( a“‘), where w6 is the vorticity corresponding to the regularized velocity (see, e.g., [ 31). Denote the A_, the integral operator that defines H-, by 11 f I)-, = l1LS f 1ILz (see Folland [8], Chapter 6). Multiplying (3.1 ) by A_, and then taking inner product with A-,( w - ah) yields

where

To bound T I , we first split TI into two parts as follows:

where A f = L , u - V f - u - VA-, f = [ A-,, u - V ] f is a commutator operator. For the Ti”term, V. u = 0 implies


According to Folland (see [8], Lemma 6.14),

(3.9)

Actually, (3.9) is not the exact statement of Lemma 6.14, but it follows from its proof because a term I 6 - rl I is needlessly replaced by ( 1 + I f - rl I 2, ' I 2 . Application of (3.9) to A = [ LS, u- V] gives

Note that IIVu + is well defined and is bounded since Vu is integrable in L2 for smooth initial vorticity with compact support. Combining (3.6)-( 3. lo), we conclude that

We now turn to estimating T 2 . Clearly we have

Note that

There are two cases. If K is a kernel of Calderon-Zygmund type (i.e., r = 1 ), then Calderon-Zygmund inequality implies that

since K6 commutes with LS. Combining the above inequality with (3.12), we have

Now if K is only L:, (i.e., r < 1 ), we would like to use Young's inequality to prove

1026 J . GOODMAN AND T. Y . HOU

(3.13). To this end, we first show that wh is of compact support. It is enough to show that uh is bounded independent of h . Define T* by

for some small E > 0 and define gx(y ) to be a smooth function satisfying gx(y) = 1 for I y - X I S 1 and gx(y) = 0 for I y - x / 2 2. We split uh into two terms as follows:

For the I1 term, K 6 ( x - y)gx(y ) is bounded. We have

by the assumption of Theorem 2. For the Z term, we express it as an inner product

Therefore, we obtain for t 5 T* that

(3.16) I ZI 5 C, + C2hP-c/62+2S 5 C ,

since 6 = h4 with q < p / ( 4 + 2s) by assumption. This proves the boundedness of uh. Consequently, oh is of compact support. Now the fact that both w and wh have compact support allows us to replace K6 in K6 * ( w - w h ) - Vw by a cut-off k6, where &(x) = K 6 ( x ) g ( x / 2 R ) , g is the cut-off function given above, and R is the radius of the support for vorticity. Since K6 is L integrable, we get by Young's inequality

II&*(O - 0h)'VfdII-S = I lA-S&*(W - wh)'VwllL2

This proves ( 3.1 3 ).


The term T3 can be bounded as TI except that we need to show

Note that for any test function 4 E L2 and 0 S J S s + 3,

By Young’s inequality, we have

where we have used llVJ+I+‘KslJL~ 5 C6-(’+I+’). This can be obtained by using the decay of K, (see, e.g., [3] ) . Thus we obtain

Consequently, we have

for t 5 T* by (3.14). It follows from (3.17)-(3.20) that

Arguing as in estimating T I and using (3.21 ), we obtain

provided that 6-(4+2s)* hs-c 5 1. This determines how much smoothing we must use. Combining (3.1 1 ), (3.13), and (3.22), we arrive at

Therefore the Gronwall inequality gives

1028 J. GOODMAN AND T. Y . HOU

for t S T*. Taking h small enough, we have from (3.24) that IIw - kJhl1-s d h S - " / 2 for t 5 T*. This implies T* = T. Consequently (3.24) is valid in the entire interval t 5 T. This completes the proof.

Remark. As we see in the proof, we do not need the assumption that oh has a bounded total vorticity in the case where K is a kernel of Calderon-Zygmund type.

Appendix

In this appendix, we prove the well-posedness of the Riesz-Euler equations for smooth initial data.

THEOREM 3. Assume that wo E Cp '( R 2 ) for 12 1 . Then there exists a unique smooth solution w E W" ', fl Cf, of the Riesz-Euler equations for a short time T and for anyp > 2.

Proof of Theorem 3: For simplicity, we only present the proof for 1 = 1 . The case of 1 L 2 can be proved similarly. Differentiating equation ( 1.4) with respect to x twice, we get

(A.1) (V2w)t + u * V(V2w) = -2vu - 0 2 0 - v2u * v w

Define the flow map X ( a, t ) as follows:

dX -(a, t ) = u ( X ( a , t ) , t ) , dt

X ( a , 0) = a . (A.2)

Integration of (A. 1 ) along characteristics from 0 to t yields

( 0 2 w ) ( x , t ) = v2wo(X- ' (x , t ) )

(A.3) (2Vu-V2w + V 2 ~ . V w ) ( X - ' ( ~ , s ) , S) d s , -6

where X- ' is the inverse mapping of X. Then incompressibility of X implies that

where the domain of integration in 11 u 11 Lp is finite since w is compactly supported. By Sobolev embedding theorem (see [ 8]), we have


Moreover,

since K* is a bounded operator in L p norm. Actually, we cannot directly apply Young's inequality in the case when K is L,'w. But the fact that w has compact support and the L p norm is defined over a bounded domain allow us to replace K by a cut-off k which is L' integrable. Now it follows from (A.4), (AS), and (A.6) that

Similar estimates apply to J ~ w ) ) L P and IIVWIILP. We get

Therefore, we conclude that (Iw I ( W ~ , p is bounded for some short time T, > 0. Applying the Sobolev embedding theorem one more time yields w E Cb.

We now turn to proving the continuous dependence of the Riesz-Euler equations on their initial data. Assume that w ( ' ) and d2) are two smooth solutions of the Riesz-Euler equations corresponding to initial values w y ) and or' respectively. Then

and

Subtracting (A.lO) from (A.9), we get

with

Since (a('), u( I ) ) and d 2 ) ) are smooth solutions, equation (A.11) can be solved by integrating along the characteristics defined by the velocity field d2). It is an easy matter to check that

1030 J. GOODMAN A N D T. Y . HOU

(A.12)

Recall that

Since K* is a bounded operator in L p norm, we conclude that

Therefore, we obtain

Similarly, we can show that

This proves the well-posedness of the Riesz-Euler equations for a short time.

Remark. Theorem 3 only guarantees short time existence. It is particularly interesting to determine whether the Riesz-Euler equations break down in a finite time.

Acknowledgements. We would like to thank J. T. Beale for some profitable discussion during the preparation of this work.

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new stability estimates for the 2-d vortex method

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