CONSERVATION PROPERTIES OF SMOOTHED PARTICLE HYDRODYNAMICS APPLIED TO THE SHALLOW WATER EQUATIONS

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1 BIT /00/ $ ?, Vol.??, No.??, pp.?????? c Swets & Zeitlinger CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS APPLIE TO THE SHALLOW WATER EQUATIONS JASON FRANK 1 an SEBASTIAN REICH 2 1 CWI, P.O. Box 94079, 1090 GB Amsteram. jason@cwi.nl 2 epartment of Mathematics, Imperial College 180 Queen s Gate, Lonon, SW7 2BZ. s.reich@ic.ac.uk Abstract. Kelvin s circulation theorem an its implications for potential vorticity (PV) conservation are among the most funamental concepts in ieal flui ynamics. In this note, we iscuss the numerical treatment of these concepts with the Smoothe Particle Hyroynamics (SPH) an relate methos. We show that SPH satisfies an exact circulation theorem in an interpolate velocity fiel, an that, when appropriately interprete, this leas to statements of conservation of PV an generalize enstrophies. We also inicate some limitations where the analogy with ieal flui ynamics breaks own. AMS subject classification: 76M28. Key wors: geophysical flui ynamics, potential vorticity conserving methos, geometric methos, smoothe particle hyroynamics. 1 Introuction an Lagrangian Equations of Motion Large scale geophysical flows in the atmosphere an ocean are incompressible an nearly two-imensional. As a result, vorticity plays a central role in geophysical flui ynamics. In the Lagrangian flui escription, conservation of vorticity an circulation follow from the fact that flui particles are ientical an thus there is a great egree of freeom in labeling particles; that is, vorticity conservation follows from the particle nature of the flui ( 2). One might therefore expect that a Lagrangian, particle-base approach woul lea to goo vorticity conservation also in a computational setting. In this article, we consier the conservation properties of the popular Smoothe Particle Hyroynamics metho (SPH) [9, 14], as briefly outline in 3. We will show that, inee, the continuous velocity fiel that interpolates the SPH particle velocities exactly satisfies Kelvin s circulation theorem ( 4). In turn, Stoke s theorem implies that absolute vorticity is also exactly conserve. Recently, Monaghan [10] has suggeste that circulation is conserve approximately by SPH ue to a iscrete relabeling symmetry. We stress that in this article we prove exact conservation of circulation for a continuous interpolate velocity fiel, which implies the convergence of the iscrete integral introuce in [10]. The Hamiltonian Particle-Mesh metho evelope by the authors in [7] inherits conservation

2 2 J. FRANK AN S. REICH of circulation from SPH, an in 5 we provie a numerical illustration of this result using the HPM metho. Aitional conservation properties of importance in geophysical flui ynamics are mass an energy conservation. Mass conservation is intrinsic to the SPH formulation. It is also well-known that SPH can be erive from a variational principle (see, e.g., [14, 3, 10]); i.e. it can be given a Hamiltonian structure, for which symplectic integrators may be use to obtain excellent energy conservation [13]. For simplicity of exposition, we consier the two-imensional shallow water equations (SWEs): (1.1) (1.2) t u = c 0 x h, t h = h x u, where u = (u, v) T is the horizontal velocity fiel, h is the layer epth, c 0 > 0 is an appropriate constant, an t = t + u x is the material time erivative. The results of this paper immeiately generalize to the rotating SWEs. This is briefly iscusse in 6. In the Lagrangian escription [12], the positions of all flui particles are given as a time epenent iffeomorphism from the flui label space A R 2 to R 2 : X = X(a, t), a = (a, b) A, X = (X, Y ) R 2. The labels are fixe for each particle, i.e. ta = 0, an the flui layer epth h is efine, as a function of the eterminant of the 2 2 Jacobian matrix through the relation (1.3) X a = (X, Y ) (a, b), h(x(a, t), t) X a = h o (a), where h o (a) is a time-inepenent function [12]. ifferentiation of (1.3) with respect to time yiels an expression that is equivalent to the continuity equation (1.2). Hence (1.3) an (1.2) are essentially equivalent statements. A natural choice for the labels a is given by a = X(0) = x which we assume from now on. Then the matrix X a is the ientity an h o the layer-epth at t = 0. Consier the integral ientity efining the layer-epth h at time t an Eulerian position x h(x, t) = h(x, t) δ(x X) X, where δ is the irac elta function. Using (1.3) we can pull this integral back to label space, arriving at the relation (1.4) h(x, t) = h o (a) δ(x X(a, t)) a,

3 CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS 3 which can be taken as the efinition of the layer epth in the Eulerian reference frame. The motion of the SWEs is given as the stationary state δl = 0 of the action integral { 1 L = h o 2Ẋ Ẋ c } 0 (1.5) 2 h a t with respect to amissible variations δx(a, t), where the ot notation refers to time ifferentiation for fixe label, i.e. the material time erivative t. The equations of motion become (1.6) t X = u (1.7) t u = c 0 X h where h is efine by (1.3) or (1.4). Since the SWEs (1.6) (1.7) are erive from a Lagrangian variational form, they are Hamiltonian, conserving the energy E = 1 h o {u u + c 0 h} a = 1 p p a + c 0 h 2 x, 2 2 h o 2 p = h o u, an the symplectic two-form ω := (p X) a [11]. 2 Vorticity Conservation in Ieal Shallow Water Flows The action integral (1.5) epens on the labels a explicitly only through the layer epth h = h o X a 1. As a result, the action is invariant to any transformation of label space (i.e. relabeling ) that leaves the eterminant X a unchange (e.g. any ivergence-free relabeling (a, b )/ (a, b) = 1 will suffice.) By Noether s theorem, this particle relabeling symmetry implies a conserve quantity of the ynamics, which turns out to be the conservation of vorticity in its many forms as outline below. Note, however, that this symmetry with implie conservation law is very much tie into the particle nature of the flow in the Lagrangian escription; it follows essentially because particles are inistinguishable from one other. It is this observation which motivates us to consier the vorticity conservation properties of Lagrangian methos such as SPH in this article. For further iscussion of the particle relabeling symmetry an its relation to vorticity, we highly recommen Salmon s monograph [12]. Consier the vorticity ζ = x u. Using x t u = x u t + ( x u)( x u) + u x ( x u) = 0,

4 4 J. FRANK AN S. REICH it is easy to conclue that vorticity satisfies the continuity equation (2.1) t ζ = ζ x u. The ratio of ζ to h, i.e., q = ζ/h, is calle the potential vorticity (PV) [12]. The PV fiel q is materially conserve since, using (1.2) an (2.1), { t q = h 1 t ζ q } t h = 0. As a result, it follows that (1.2) an (2.1) are special cases of an infinite family of continuity equations (2.2) t [hf(q)] = x [hf(q)u], where f is an arbitrary (smooth) function of q. Let us now iscuss the concept of circulation. Take a close loop S = {a(s)} s S 1 in label space an consier the particle locations X(s) = X(a(s)) parameterize by s S 1. By efinition, the loop {X(s)} s S 1 in configuration space is avecte along the velocity fiel, i.e. X(s) = u(x(s)). t Kelvin s circulation theorem [12] states that (2.3) u X s s = 0. t Inee, we obtain ( ) ( ) u X s s = t t u X s s + u s t X s = c 0 X h X s s + u u s s ( ) 1 = 2 (u u) s c 0 h s s = 0. Let V enote the area enclose by S in label space an R its image in x-space. Then Stokes theorem applie to (2.3) yiels X u) X a a = t V( (2.4) ( x u) x = 0. t R Because V is arbitrary, the left sie of this equation yiels another statement of PV conservation, since ( X u) X a = h o ζ h = h oq, an h o /t = 0. Similarly, after applying the transport theorem [5] to the right equality in (2.4), a secon appeal to the arbitrariness of V yiels the vorticity equation (2.1).

5 CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS 5 3 Review of Smoothe Particle Hyroynamics The Smoothe Particle Hyroynamics (SPH) metho [9] is a Lagrangian metho for flui ynamics, in which the flui mass is istribute over a number of smooth, compactly supporte, particle-centere basis functions. A similar metho is the Finite Mass Metho [14]. Assume that a set of Lagrangian particles with positions {X k (t); X k R 2 } is given as a function of time an that (3.1) t X k = u k, where u k is the velocity of the particle. Then the time evolution of a quantity g, satisfying a continuity equation can be approximate by t g = x [gu], ḡ(x, t) = k γ k ψ(x X k (t)). Here {γ k } are constants etermine by the initial g(x) fiel an ψ is an appropriate basis function. Typically, SPH is implemente with raially symmetric basis functions, i.e. ψ(x) = Ψ( x ) an Ψ(r) is either a Gaussian, a compactly supporte raial basis function [4], or a raial spline [9]. Let us apply this iea to the layer-epth h, i.e., we introuce the approximation (3.2) h(x, t) = k m k ψ(x X k (t)) an assume that h(x, t) > 0. Then each particle contributes the fraction (3.3) ρ k (x, t) := m kψ(x X k (t)) h(x, t) to the total layer-epth. These fractions form a partition of unity, i.e. ρ k (x, t) = 1. k Hence they can be use to approximate ata from the particle locations to any x R 2. In particular, we efine an approximate Eulerian velocity fiel (3.4) ū(x, t) := k ρ k (x, t) u k (t) with layer epth flux ensity (inserting (3.3)) h(x, t) ū(x, t) = k m k ψ(x X k (t)) u k (t).

6 6 J. FRANK AN S. REICH Using (3.1), it is now easily verifie that (3.5) t h(x, t) + x [ h(x, t)ū(x, t)] = 0. It follows that the layer epth approximation (3.2) exactly satisfies the continuity equation (1.2) uner the flow of the formally efine velocity fiel (3.4). In general the particle avection velocity is ifferent from the approximate velocity, i.e u k ū(x k ). We note that the moification suggeste in [8] to avoi penetration in SPH correspons 1 to avecting the particles in the velocity fiel (3.4). Hence, (3.1) an (3.2) provie an approximation to the continuity equation (1.2). To get a close system of iscretize equations, we still have to approximate the momentum equation (1.1). For example, one can use (3.6) t u k = c 0 x h(x, t) x=xk = c 0 m j Xk ψ(x k X j ). The equations (3.1), (3.2), an (3.6) comprise the stanar SPH approximation to the SWEs (1.1) (1.2). We introuce the canonical momenta p k = m k u k. The equations (3.1), (3.2), an (3.6) are now canonical with Hamiltonian (energy) (3.7) H = 1 2 k 1 p k 2 + c 0 m k 2 j m k m l ψ(x l X k ) an symplectic structure ω = k p k X k. A numerical time-stepping scheme is obtaine by noting that t X k = u k, l,k t u k = 0 can be solve exactly an that the implie time evolution of h(x, t) exactly satisfies (3.5). Similarly, equation (3.6) an t X k = 0 can also be integrate exactly since in this case h t = 0. A composition of these exact propagators leas to a symplectic time-stepping scheme [13] implying goo long-time energy conservation [2]. Bonet & Lok [3] have also iscusse conservation properties of SPH in a variational formulation. In particular they show conservation of linear an angular momentum, provie the basis function ψ is raially symmetric. These follow from the fact that the Hamiltonian (3.7) is invariant uner translations an rotations in the Lagrangian particle positions {X k } an subsequent application of Noether s theorem [11]. 1 More precisely, the formulation Eqn. (2.6) in [8] avocates particle avection in an approximate velocity fiel base on a generic kernel. Taking this kernel to be ψ yiels the avection fiel ū( ).

7 CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS 7 4 Vorticity Conservation Properties of SPH Circulation is conserve, using SPH, for an interpolate velocity fiel u(x) efine as follows: at time t = 0, let u(x) be any initial velocity fiel satisfying u(x k ) = u k (0) at the particle locations X k. (For example, suppose the particle velocities at t = 0 are given as a continuous function.) efine natural labels a = X(0) = x an let the fiel of particle locations X(a) an particle velocities u(a) = u(x(a)) evolve uner the solution of the SPH flow ue to (3.6) accoring to the orinary ifferential equations (i.e. ecouple in label space) (4.1) t u = c 0 X h(x, t) = c0 m k X ψ( X X k (t) ), k t X = u. Note that for this velocity fiel it oes hol that u(x k ) = u k for all time t, in contrast to the approximate velocity fiel ū(x) of the continuity equation (3.4). Figure 4.1 illustrates the relationship between the velocity fiels u an ū. S u(x k ) = uk X(s) u(x(s)) X k u(x ) k Figure 4.1: A close curve X(s) avecte with the flow, illustrating the velocity fiels u an ū. The SPH particle at X k (support inicate by the otte line) with velocity u k remains on the curve throughout the integration. Now, we efine a curve of Lagrangian points X(s) = X(a(s)) with s S 1 an S = {a(s)} s S 1 being a close loop in label space. The associate loop {X(s)} s S 1 in configuration space is propagate in the velocity fiel u(s) = u(x(s)) accoring to X(s) = u(s). t

8 8 J. FRANK AN S. REICH We assume that X(s) an u(s) are sufficiently ifferentiable. circulation theorem (2.3) becomes (4.2) u X s s = 0. t Inee, we obtain ( ) ( ) u X s s = t t u X s s + u s t X s = c 0 X h Xs s + u u s s ( ) 1 = 2 (u u) s c 0 hs s = 0. Then Kelvin s It is important to notice that the circulation theorem above inuces a true constraint on the numerical solution, since any SPH particle that is initially locate on the loop S will remain on the loop as the integration procees, an furthermore, the particle velocity u k will be exactly interpolate by u(x k ). If we now give each particle insie S a label a an let V enote the area enclose by S an let R enote the image of V in x-space, then applying Stokes theorem to (4.2) yiels (4.3) t for which the left sie implies (4.4) V( X u) X a a = t R t {( X u) X a } = 0, ( x u) x = 0, since V is arbitrary. Let h(x(a)) enote the layer-epth approximation obtaine as the solution of the continuity equation (1.2) along the interpolate velocity fiel u(x(a)). Then h X a = h o an equation (4.4) implies conservation of the PV fiel q = ( X u)/h, i.e. q/t = 0. Furthermore, applying the transport theorem to the right equality of (4.3), an again noting that V is arbitrary, yiels a continuity equation for the absolute vorticity of the velocity fiel u: t ζ = x (ζu), ζ = x u, cf. (2.1). We woul a that (4.2) an (4.3) are preserve uner time iscretization via a splitting as escribe in the previous section. For a numerical verification of (4.2) one can represent the loop {X(s)} by a sufficient number of particles { ˆX l } with associate velocities {û l } satisfying

9 CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS 9 (4.1). Note that the particles { ˆX l } may be compute along with the SPH simulation as passive tracers with zero mass. The integral (4.2) is approximate by (4.5) u X s s û l ( ˆX l+1 ˆX l ). l One shoul observe that the variation of this integral in time converges to zero as the number of points iscretizing the loop is increase. Such a verification is inclue in 5 for the Hamiltonian Particle-Mesh metho. Monaghan [10] motivates the approximate conservation of (4.5), appealing to Noether s theorem for a iscrete relabeling (change of inex) of particles of equal mass. This reasoning is limite because Noether s theorem applies only to continuous symmetries. However, our result shows that the approximate conservation of (4.5) is an implication of a stronger result, namely the exact conservation of (4.2). We have seen that the PV fiel q = ( X u)/h is exactly conserve in SPH. It is also easy to verify that the prouct hf(q), where f is an arbitrary function of q, exactly satisfies the continuity equation (1.2) in the interpolate velocity fiel u(x). For iagnostic purposes, one can obtain a computable continuous approximation of hf(q) by again taking a sufficient number of passive tracer particles { ˆX l } with associate velocities {û l } satisfying (4.1). Following the iscussion for the layer-epth h in 3, we efine a continuous approximation hf(q)(x, t) = l α l ψ(x ˆX l (t). Given a particle mass m l an PV value q l for each tracer particle ˆX l, as etermine by the initial ata, the weights {α l } can be efine by α l = m l f(q l ). It is straightforwar to show that hf(q) satisfies a moifie continuity equation of type (3.5). Uner perioic bounary conitions, this continuity equation implies the exact conservation of the generalize enstrophies Q f = hf(q) x. Since this is also true for the split equations of motion use for the time-stepping, the overall space-time approximation conserves enstrophy. The number of tracer particles can be chosen to be quite large if, for example, the statistical/spectral properties of the SPH approximation to hq 2 are of interest. 5 Hamiltonian Particle-Mesh Metho an Numerical Verification The SPH equations of motion are equivalent to the simulation of a molecular flui with a softene repulsive pair potential given by the SPH basis function ψ(x) = Ψ( x ) [1]. In general, such flows ten to a statistically uniform state of isorer. In practice, therefore, SPH is use with some form of artificial viscosity, an this results in a monotone ecrease in circulation along any loop, an a monotone loss of energy.

10 10 J. FRANK AN S. REICH To improve the stability of SPH, we suggeste in [7] working with an average SWE (5.1) t X = u, (5.2) t u = c 0 X (A h), where A is some smoothing operator. These equations are still canonical with Hamiltonian H = 1 p p a + c 0 (5.3) h(a h) x 2 h o 2 an circulation preserving. Let {X k } be an initially equi-istribute set of points with an associate area A. Numerically the layer-epth h is now approximate by the singular measure h(x, t) = k m k δ(x X k (t)), m k = h o (a k ) A, a k = X k (0), which approximates the integral (1.4) in a weak sense. If, for example, A is chosen to be convolution with the SPH basis function ψ such that (A h) (x, t) = h(x, t) = k m k ψ(x X k (t)), then the layer-epth epenent part of the Hamiltonian (5.3) becomes c 0 h(a 2 h) x = c 0 h h x = c 0 m k m l ψ(x k X l ) 2 2 an the stanar SPH metho is recovere. However, typical SPH basis functions ψ o not provie enough smoothing, which results in the above mentione tenency to a state of isorer. The layer-epth approximation h satisfies the continuity equation (1.2) in a weak sense. Furthermore, the following integral version of (1.3) is easily shown: h X a a = h x = V R l,k k:x k R m k an m k h o (a) a. k:x k R V The set of particles over which the sum is performe is constant provie no particle enters or leaves the omain R which woul correspon to a singular X a an, hence, to a non-physical state.

11 CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS 11 The Hamiltonian Particle-Mesh metho (HPM), introuce in [7], iffers from SPH primarily in the construction of the smoothe layer epth A h. Specifically, we efine a uniform gri with gri points x ij an gri spacing x. Let h SPH (x, t) be the SPH approximation to the layer epth (3.2) with the SPH basis function replace by a tensor prouct basis function ψ(x X k ) := φ( x X k )φ( y Y k ), where φ(r) is given by the cubic spline 2 3 r r3, r R 1 φ(r) = 6 (2 r)3, R < r 2R 0, r > 2R for R = x. efine the grie layer epth values h ij SPH (t) = h SPH (x ij, t) an let the matrix S = {Sij mn } enote the representation of a spatial averaging operator S over the given gri {x ij }. Since the cubic splines form a partition of unity on the gri, we can efine a continuous approximation of a smoothe/average layer-epth in space (A h) (x, t) = h HPM (x, t) = ij,mn h ij SPH (t) Smn ij ψ(x mn x). This approximation, use in the HPM metho, can be viewe as a spatial averaging over short wave-length isturbances in stanar SPH. One can also think of the HPM metho as an efficient implementation of the SPH metho for a globally supporte basis function ψ efine by A δ = ψ := S ψ, ψ a stanar SPH basis function. For a more etaile escription of the HPM metho, incluing its Hamiltonian structure, see [7]. The HPM metho conserves circulation using the same proof as for SPH in 4. The essential observation is that particles are avecte in a velocity fiel that exactly evolves in some continuous approximate layer epth. We have performe an experiment with HPM to verify the conservation of circulation. The flow moels the interaction of two positively oriente vortices in a rotating reference plane. We iscretize this flow using HPM on a gri with particles. We intentionally chose a fairly coarse iscretization for this problem to illustrate that the circulation theorem hols inepenent of the precision of the iscretization. Initially a circular loop of M evenly space particles of zero mass was place in the flow. The solution incluing the loop particles was evolve over time intervals of T = 3, T = 6 an T = 15 revolutions of the plane. The experiments

12 12 J. FRANK AN S. REICH were repeate, each time refining the iscretization, for M = 100, 200, 400, 800 an 1600 particles. The circulation integral was approximate using (cf. (4.5)) C M (t) = m û m ( ˆX m+1 ˆX m 1 ) Figure 5.1 shows the eformation of the loop at time T = 15, compute using M = 3200 particles. The loop is superimpose over a contour plot of potential vorticity, an its interior is shae. In Figure 5.2 we see secon orer convergence of (C M (t) C M (0))/C M (0) to zero as M increases. Convergence of this sum as M is implie by (4.2) Figure 5.1: Final eformation of the circular loop at time T = 15, using M = 3200 particles. 6 Concluing Remarks In this article, we have shown that the SPH metho with (3.6) satisfies a Kelvin circulation law. The results are base on the introuction of a continuous velocity fiel u(x) which interpolates the particle velocities u k for all time

13 CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS T=15 (C M (T) C M (0))/C M (0) T=3 T= M Figure 5.2: Convergence of the iscretization of the circulation integral. an is avecte in the flow of the continuous SPH approximation (4.1). This velocity fiel conserves circulation (4.2) an, by Stokes theorem, absolute vorticity (4.3). Furthermore, we can formally efine a layer epth approximation h such that h satisfies the continuity equation (1.2) in the interpolate velocity fiel. efining the potential vorticity with respect to this layer epth yiels exact PV conservation. One can expect a particle metho to converge to the solutions of the average SWEs (5.1)-(5.2) for an appropriate smoothing operator A an in the limit of large particle numbers. The necessary amount of regularity an the impact of the smoothing operator A on the long term ynamics of the SWEs are not yet clear. The results of this paper easily generalize to the rotating SWEs t u = f 0u c 0 x h, t h = h x u, where u = ( v, u) T an f 0 /2 is the angular velocity of the reference plane. Potential vorticity is now efine by q = x u + f 0 h

14 14 J. FRANK AN S. REICH an Kelvin s circulation theorem becomes 0 = ( u + f ) 0 t 2 X X s s = ( X u + f 0 ) X a a t V = ( x u + f 0 ) x. t R We wish to mention the Balance Particle-Mesh (BPM) metho of [6] which uses raial basis functions to approximate the absolute vorticity ω = x u + f 0. See [6] for the geometric properties of the BPM metho. Kelvin s circulation theorem also applies to three-imensional ieal fluis while conservation of PV takes a more complicate form (see [12]). Again, conservation of circulation can be shown for the SPH metho in the same manner as outline in this note for two-imensional fluis. In fact, the concept of circulation even applies to molecular simulations of a mono-atomic liqui [1] with Hamiltonian H = 1 2m p k 2 + φ( X k X l ), l>k k where m is the atomic mass an φ(r) an interaction potential. We introuce the function ρ(x, t) = l an note that Newton s law is equivalent to φ( x X l (t) ), x R 3, t p k = Xk H = x ρ(x = X k, t). We also have t X k = 1 m p k. Applying the notations of 4 an efining u = p/m, we obtain the circulation theorem u X s s = 0 t an, in two imensions, conservation of vorticity per control area, i.e., ( x u) x. t R One shoul keep in min that φ(r) is often singular at r = 0 an, hence, ρ(x, t) is not efine for x = X k. However, one can replace φ(r) by a smooth truncation φ(r) such that φ(r) = φ(r) for r r o an ρ (0) = 0, ρ(0) <. Here r o is chosen such that X i (t) X j (t) > r o for all t 0 an all i j. One shoul also note that, contrary to flui ynamics, the prouct ρ X a nee not to be approximately conserve an that X a can become singular.

15 CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS 15 Acknowlegements. J. Frank gratefully acknowleges partial funing by GM, Bonn. Partial financial support of S. Reich by EPSRC Grant GR/R09565/01 an by European Commission funing for the Research Training Network Mechanics an Symmetry in Europe is gratefully acknowlege. REFERENCES 1. M. P. Allen &. J. Tilesley, Computer Simulation of Liquis, Oxfor University Press, Oxfor, G. Benettin & A. Giorgilli, On the Hamiltonian interpolation of near to the ientity symplectic mappings with application to symplectic integration algorithms, J. Stat. Phys., 74 (1994) J. Bonet & T.-S.L. Lok, Variational an momentum preservation aspects of smooth particle hyroynamic formulations, Comput. Methos Appl. Mech. Engrg., 180 (1999) M.. Buhmann, Raial basis functions, Acta Numerica, (2000) A. J. Chorin & J. E. Marsen, A Mathematical Introuction to Flui Mechanics, Springer-Verlag, New York, J. Frank & S. Reich, A particle-mesh metho for the shallow water equations near geostrophic balance, J. Sci. Comput., to appear. 7. J. Frank, G. Gottwal & S. Reich, The Hamiltonian particle-mesh metho, Meshfree Methos for Partial ifferential Equations, Lecture Notes in Computational Science an Engineering, Vol. 26, Springer, J. J. Monaghan, On the problem of penetration in particle methos, J. Comput. Phys., 82 (1989) J. J. Monaghan, Smoothe particle hyroynamics, Ann. Rev. Astron. Astrophys., 30 (1992) J. J. Monaghan, SPH an the alpha turbulence moel, preprint, P. J. Morrison, Hamiltonian escription of the ieal flui, Rev. Moern Phys., 70 (1998) no. 2, R. Salmon, Lectures on Geophysical Flui ynamics, Oxfor University Press, Oxfor, J. M. Sanz-Serna & M. P. Calvo, Numerical Hamiltonian Problems, Chapman an Hall, Lonon, H. Yserentant, A new class of particle methos., Numer. Math., 76 (1997)