Centrum voor Wiskunde en Informatica

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1 Centrum voor Wiskune en Informatica Moelling, Analysis an Simulation Moelling, Analysis an Simulation Conservation properties of smoothe particle hyroynamics applie to the shallow water equations J.E. Frank, S. Reich REPORT MAS-R11 AUGUST 20

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3 Conservation Properties of Smoothe Particle Hyroynamics Applie to the Shallow Water Equations Jason Frank CWI P.O. Box 94079, 1090 GB Amsteram, The Netherlans Sebastian Reich epartment of Mathematics, Imperial College 180 Queen s Gate, Lonon, SW7 2BZ, UK s.reich@ic.ac.uk ABSTRACT Conservation of potential vorticity (PV) an Kelvin s circulation theorem express two of the funamental concepts in ieal flui ynamics. In this note, we iscuss these two concepts in the context of the Smoothe Particle Hyroynamics (SPH) metho. We show that, if interprete in an appropriate way, one can make a statement about conservation of circulation an PV in a particle flui moel such as SPH. We also inicate some limitations where the analogy with ieal flui ynamics breaks own Mathematics Subject Classification: 76M28 Keywors an Phrases: geophysical flui ynamics, potential vorticity conserving methos, geometric methos, smoothe particle hyroynamics Note: Work of J. Frank carrie out uner project MAS Atmospheric Flow an Transport Problems. Partial support by GM is gratefully acknowlege. Note: Partial financial support of S. Reich by EPSRC Grant GR/R09565/ an by European Commission funing for the Research Training Network Mechanics an Symmetry in Europe is gratefully acknowlege. 1. Introuction Smoothe Particle Hyroynamics (SPH) [7, 10] is a popular particle base metho for computational flui ynamics. It is well-known that the SPH metho can be erive from a variational principle (see, e.g., [10]) or, in other wors, can be given a Hamiltonian structure. In this note, we aress the important question of conservation of potential vorticity an circulation in SPH simulations. These conserve quantities follow from the unerlying relabeling symmetry of ieal flui ynamics an are funamental to the long-time solution behavior [8]. For simplicity of exposition, we consier the two-imensional shallow water equations (SWEs): t u = c 0 x h, (1.1) t h = h x u, (1.2) 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. 2. Conserve Quantities in Ieal Shallow Water Flows The SWEs (1.1)-(1.2) possess a number of conserve quantities which are important for the longtime ynamics. Let us start with the vorticity ζ = x u. Using x t u = x u t +( x u)( x u)+u x ( x u)=0,

4 2. Conserve Quantities in Ieal Shallow Water Flows 2 it is easy to conclue that vorticity satisfies the continuity equation t ζ = ζ x u. (2.1) The ratio of ζ to h, i.e., q = ζ/h, is calle the potential vorticity (PV) [8]. The PV fiel q is materially conserve since { t q = h 1 t ζ q } t h =0. Using this result, it follows that (1.2) an (2.1) are special cases of an infinite family of continuity equations t [hf(q)] = x [hf(q)u], (2.2) where f is an arbitrary (smooth) function of q. Next we introuce flui particle labels a =(a,b) R 2 an efine particle positions x(a,t) R 2 via a iffeomorphism. The labels are fixe for each particle [8], i.e. t a = 0. The eterminant of the 2 2 Jacobian matrix satisfies x a = (x, y) (a, b) h x a = h o, where h o (a) is a time-inepenent function [8]. 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. t x(s)=u(x(s)). Kelvin s circulation theorem [8] states that u x s s =0. (2.3) 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 A 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 b = t A( ( x u)x y =0. (2.4) t R

5 3. Raial Basis Functions an Smoothe Particle Hyroynamics 3 Since A 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 [4] to the right equality in (2.4), a secon appeal to the arbitaryness of A yiels the vorticity equation (2.1). The SWEs also conserve energy E = 1 h o {u u + c 0 h} a b = 1 h {u u + c 0 h} x y 2 2 an the symplectic two-form ω := (u x)a b [8]. 3. Raial Basis Functions an Smoothe Particle Hyroynamics 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 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 t g = x [gu], can be approximate by (3.1) g(x,t)= k γ k ψ( x X k (t) 2 ). Here {γ k } are constants etermine by the initial g(x)fielanψ(r) is an appropriate raial basis function [3], for example, ψ(r 2 )= ( ( r ) ) 2 1/2 + c 2, R c, R > 0 two parameters. Let us apply this iea to the layer-epth h, i.e., we introuce the approximation h(x,t)= k m k ψ( x X k (t) 2 ) (3.2) an assume that h(x,t) > 0. Then each particle contributes the fraction ρ k (x,t):= m kψ( x X k (t) 2 ) h(x,t) (3.3) 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 interpolate ata from the particle locations to any x R 2.Inparticular, we efine an interpolate Eulerian velocity fiel u(x,t):= k ρ k (x,t)u k (t) (3.4)

6 4. Conservation of Enstrophy an Circulation 4 an a layer epth flux ensity h(x,t)u(x,t)= k m k ψ( x X k (t) 2 )U k (t). Using (3.1), it is now easily verifie that t h(x,t)+ x [h(x,t)u(x,t)] = 0. (3.5) 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 interpolate velocity, i.e U k u(x k ). We note that the moification suggeste in [6] 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 t U k = c 0 x h(x = X k,t)= c 0 m j Xk ψ( X k X j 2 ). (3.6) j The equations (3.1), (3.2), an (3.6) are commonly known as a smoothe particle hyroynamics (SPH) approximation [7] to the SWEs (1.1)-(1.2). A variant of these equations is obtaine by replacing (3.6) by t U k = c 0 2 (m k + m j ) Xk ψ( X k X j 2 ). (3.7) j The equations (3.1), (3.2), an (3.7) are now canonical with Hamiltonian (energy) H = 1 2 k U k 2 + c 0 2 m k ψ( X l X k 2 ) l,k an symplectic structure ω = k U k X k. If we assume that the particle locations {X k } have been chosen such that m k = m = const., then (3.6) an (3.7) coincie. A numerical time-stepping scheme is obtaine by noting that t X k = U k, t U k = 0 can be solve exactly an that the associate time evolution of h(x,t) exactly satisfies (3.5). Similarly, equation (3.7) an t X k = 0 can also be integrate exactly since h t = 0. A composition of these exact propagators leas to a symplectic time-stepping scheme [9] implying goo long-time energy conservation [2]. The same approach using (3.6) can be mae time-symmetric but is not symplectic, in general. 4. Conservation of Enstrophy an Circulation The PV fiel q is materially conserve. This suggests assigning a constant PV value q k to each particle location X k, thereby trivially enforcing the PV conservation law q/t =0. Usingthe fractions ρ k, we obtain the interpolate fiels f(q(x,t)) := k f(q k )ρ k (x,t), (4.1) 1 More precisely, the formulation Eqn. (2.6) in [6] avocates particle avection in an interpolate velocity fiel base on a generic kernel. Taking this kernel to be ψ yiels the avection fiel u(x k ).

7 4. Conservation of Enstrophy an Circulation 5 where f is again an arbitrary function of q. It is easily verifie that the approximation (4.1) satisfies the continuity equation { } t [hf(q)](x,t)= x m k f(q k )ψ( x X k (t) 2 )U k k corresponing to (2.2). Uner the given perioic bounary conitions, this iscrete conservation law implies the exact conservation of the generalize enstrophies Q f = hf(q)x y. Since this is also true for the split equations of motion use for the time-stepping, the overall space-time iscretization conserves enstrophy inepenently of whether (3.6) or (3.7) is use. One shoul, however, be aware that the prouct hq is not equivalent to the vorticity of the interpolate velocity fiel u as is true for the SWEs. See below. The circulation is also conserve, but for a ifferent velocity fiel U(x,t) efine as follows: Let U(x,0) be any initial velocity fiel satisfying U(X k,0) = U k (0) at the particle locations. (For example, suppose the particle velocities at t = 0 are given as a continuous function.) An let U(x,t) evolve uner the solution of the SPH flow ue to (3.6) accoring to t U(x)= c 0 X h(x,t)= c 0 m k x ψ( x X k (t) 2 ). k Note that for this velocity fiel it oes hol that U(X k (t)) = U k (t). Figure 1 illustrates the relationship between the velocity fiels u an U. X k U k u(x(s)) u(x k ) S X(s) U(s) X l u(x l) U l Figure 1: A close curve avecte with the flow, illustrating the velocity fiels u an U. 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(x,t) accoring to t X(s)=U(X(s)).

8 4. Conservation of Enstrophy an Circulation 6 We assume that X(s)anU(s) are sufficiently ifferentiable. Then Kelvin s circulation theorem (2.3) becomes U X s s =0. (4.2) 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. This reasoning is no longer vali if the momentum equation (3.7) is use. Conservation of circulation is thus in conflict with conservation of symplectic structure. If we now give eachparticle insie S alabela an let A enote the area enclose by S an let R enote the image of A in x-space, then applying Stokes theorem to (4.2) yiels t A( x U) X a a b = t for which the left sie implies R ( x U)x y =0, (4.3) t {( x U) X a } =0, (4.4) since A is arbitrary. Note that, in principle, the eterminant X a can be compute along a given particle path X k (t) by integrating the linearize SPH equations along X k (t). However, the prouct of X a an h is not equivalent to a time-inepenent function h o (a) an, hence, equation (4.4) is not equivalent to conservation of PV in the stanar sense, i.e. q/t =0. Applying the transport theorem to the right equality of (4.3), an again noting that A is arbitrary, yiels a continuity equation for the relative vorticity of the velocity fiel U: ζ t = x (ζu), ζ = x U, cf. (2.1). However, the vorticity ζ(x,t) will be ifferent from the quantity h(x,t)q(x,t)= k q k m k ψ( x X k (t) 2 ), even if the two are initially equal, because the latter is avecte by the interpolate velocity fiel u. Hence the agreement between u an U is a funamental measure of the quality of an SPH simulation. 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 has to represent the loop {X(s)} by a sufficient number of particles { X l } with associate velocities {Ū l }. The integral (4.2) is then approximate by U X s s Ū l ( X l+1 X l ). l Note that the particles { X l } are not part of the SPH iscretization but are instea obtaine via post-processing of the SPH solutions.

9 5. Concluing Remarks 7 5. Concluing Remarks In this note, we have shown that the SPH metho with (3.6) satisfies a Kelvin circulation law. However, conservation of circulation is in conflict with conservation of symplecticness unless the weights m k are all equal. We have also shown that conservation of circulation implies a form of PV conservation an that the generalize enstrophies Q f are also preserve. Note that the numerical approximations o not, in general, satisfy the funamental relations ζ = hq an h x a = h o.the consequences of this limitation require further investigations. 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 an Kelvin s circulation theorem becomes t ( u + f 0 2 x ) x s s = t A ( x u + f 0 ) x a a b = t R ( x u + f 0 ) x y =0. We wish to mention the Balance Particle-Mesh (BPM) metho of [5] which uses raial basis functions to approximate the absolute vorticity ω = x u + f 0. See [5] 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 [8]). 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 P k 2 + φ( X k X l ), 2m l>k k where m is the atomic mass an φ(r) an interaction potential. We introuce the function ρ(x,t)= l φ( x X l (t) ), x R 3, an note that Newton s law is equivalent to 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 replacing U by P, we obtain the circulation theorem t P X s s =0 an, in two imensions, conservation of vorticity per control area, i.e., ( x P)x y. 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) <. Herer o is chosen such that X i (t) X j (t) >r o for all t 0 an all i j.

10 References 8 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), 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, submitte. 6. 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), 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,