N-Body Simulation. Typical uncertainty: π = 4 Acircle/Asquare! 4 ncircle/n. πest π = O(n

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1 N-Body Smulaton Solvng the CBE wth a 6-D grd takes too many cells. Instead, we use a Monte-Carlo method. Example: Monte-Carlo calculaton of π. Scatter n ponts n square; count number ncrcle fallng wthn crcle. π = 4 Acrcle/Asquare! 4 ncrcle/n Typcal uncertanty: πest π = O(n 1/2 )

2 Representng the Dstrbuton Functon Replace smooth dstrbuton wth bodes: f ( r, v) {(m, r, v ) = 1,...,N} f ( r, v) Ths requres that for any phase-space volume V, Z d rd v f ( r, v) = m ( r, v ) V E.g., select ( r, v ) wth probablty proportonal to f ( r, v ), and assgn all bodes equal mass: V N m δ 3 ( r r )δ 3 ( v v ) =1 m = 1 N Z d rd v f ( r, v)

3 Advancng Tme Move bodes along phase flow (method of characterstcs): Estmate potental from N-body representaton: Ths wll yeld the usual N-body equaton for pont masses. But sngular potentals are awkward, so we smooth the densty feld: δ 3 ( r r ) 3 4π ε 2 ( r r 2 + ε 2 ) 5/2 Ths substtuton yelds the followng equatons of moton: d r dt = v ( r, v ) = ( v, ( Φ) ) 2 Φ r = 4πG d v dt = N j N m δ 3 ( r r ) =1 Gm j ( r j r ) ( r j r 2 + ε 2 ) 3/2 Plummer (1911) smoothng Aarseth (1963)

4 Comments 1. N-body models relax at roughly the same rate as real stellar systems wth the same N ; the relaxaton tme s t r N 8ln(R /ε) t c 2. Samplng proportonal to f ( r, v ) s the smplest opton, but not the only one; other weghtng schemes are also possble. E.g., use dfferent masses when samplng f s ( r, v) and f d ( r, v), or make m depend on f ( r, v ). But note effect on relaxaton tme! 3. Plummer smoothng s just one of many possbltes, and may not be optmal; one alternatve wth less of a tal s δ 3 ( r r ) 15 ε 4 8π ( r r 2 + ε 2 ) 7/2 Dehnen (2001)

5 Force Calculaton: Drect Summaton Smplest method: sum over all other bodes. a = N j Gm j ( r j r ) ( r j r 2 + ε 2 ) 3/2 Advantages: robust, accurate, completely general. Dsadvantage: computatonal cost per body s O(N) ; need O(N 2 ) operatons to compute forces on all bodes. However, drect summaton s a good ft wth 1. Indvdual tmesteps (see Sverre s lectures) 2. Specalzed hardware (see Smon s lectures)

6 Tree Codes Long-range gravtatonal feld domnated by monopole term: φ Gm r + O(r 3 ) Dvde system nto herarchy (.e. tree ) of compact cells: Saul Stenberg, A vew of the World from 9 th Avenue, Barnes & Hut (1986) Replace sum over N bodes wth sum over N c O(logN) cells; cost to fnd forces on all bodes s O(N logn). N N c c

7 Tree Codes Contnued 1. To compute potental at r due to a cell c : a) f c s too close to r, sum the potentals of ts sub-cells; b) otherwse, approxmate the potental as Gm c / r r c. 2. too close can be defned n varous ways; e.g.: cell s center of mass a) geometrcally: r r c < l c /θ (BH86) or < l c /θ + δ c (B95), b) dynamcally: r r c 4 < GM c l 2 c/α a (GADGET-2: Sprngel 2005). 3. Dfferent tree structures gve roughly equvalent results: a) oct-trees: 1 cube 8 cubes (Barnes & Hut 1986), b) kd trees: dvde at medan along x,y,z (Dkaakos & Stadel 1996), c) partcle trees: group nearest neghbors (Appel 1985, Press 1986). 4. Hgher moments mprove accuracy: a) quadrupole potental term: 1 2Gδ r Q c δ r /δr 5 (Hernqust 1987), b) source/snk symmet. momentum cons., O(N) (Dehnen 2000).

8 Self-Consstent Feld Method Represent potental and densty as sums: Φ( r) = k A k Φ k ( r), ρ( r) = k A k ρ k ( r) where are coeffcents and the bass functons and are A k b-orthogonal and satsfy the PE: I k δ k k = Z d r ρ k ( r)[φ k ( r)], The coeffcents are computed usng overlap ntegrals: A k = 1 Z d r ρ( r)[φ k ( r)] = 1 m [Φ k ( r )] I k I k Φ k 2 Φ k = 4πGρ k The cost of calculatng forces on all bodes s just O(N). ρ k Wth the rght bass set, SCFM yelds good forces for spherodal systems wth only a few terms (Hernqust & Ostrker 1992).

9 Tme Step Algorthms The underlyng symmetry of the N-body equaton of moton becomes evdent n Hamltonan formulaton: d r dt = H p, d p dt = H r Ths symmetry has mportant consequences: 1. dynamcal evoluton conserves phase space volume 2. Hamltonan systems have no attractors 3. dynamcal evoluton s reversble. Hamltonan systems are not structurally stable; most ntegrators wll not preserve these propertes. An ntegraton algorthm wth Hamltonan symmetry s desrable; such an ntegrator s known as symplectc.

10 Leapfrog Integrator Ths very smple ntegrator explctly preserves the symmetry of the Hamltonan equatons of moton: r [k+1] v [k+3/2] v [k+1/2] = r [k] + t v [k+1/2] = v [k+1/2] + t a ( r [k+1] ) In addton, t s accurate to second order, meanng that the error s O( t 3 ) per step, and requres very lttle storage. A drawback of the leapfrog s that veloctes are a half-step out of sync wth postons; ths can be avoded as follows: r [k+1] v [k+1] = v [k] = r [k] = v [k+1/2] + t 2 a ( r [k] ) + t v [k+1/2] + t 2 a ( r [k+1] ) Ths formulaton ntroduces a one-tme error of otherwse equvalent to the standard leapfrog. O( t 2 ) but s

11 Indvdual Tme Steps? Leapfrog becomes naccurate f t s not constant and dentcal for all bodes. Ths seems neffcent. However, the symplectc propertes of the leapfrog gve t much more stablty than most ntegrators. Algorthms n whch t s determned by current condtons are not reversble. Symmetrzng the tme step between endponts t and t + t works but mposes a sgnfcant overhead (Hut et al. 1995). A 4 th order symplectc scheme allowng ndvdual and adaptve tme-steps s now avalable (Farr & Bertschnger 2007). Sprngel (2005)

12 Errors and Relaxaton N-body smulatons dverge from exact solutons of CBE and PE for several reasons: 1. Roundoff errors. 2. Truncaton n tme steppng. 3. Force calculaton approxmatons. 4. Densty feld smoothng. 5. Relaxaton due to fnte N. An Introducton to Error Analyss, John R. Taylor In theory, these effects can all be controlled at a prce. Error #1 s seldom an ssue, whle errors #2 and #3 are easly lmted. Smoothng and relaxaton are harder to balance; hgh resoluton demands shorter tmesteps and more bodes. Fnally, all N-body systems wth N > 2 are potentally chaotc, whle the role of chaos n real galaxes s unclear.

13 Parameter Choces The number of bodes N s the key parameter: 2-body relaxaton tme: t r Nt c /8ln(R /ε) Monte-Carlo errors: O(N 1/2 ) Maxmum duraton of smulaton must be. t t r Typcal errors n acceleraton should be δa/a N 1/2. Global energy should be conserved to δe/e N 1/2. Smoothng length s typcally R N 1/2 < ε < R N 1/3. Leapfrog tme-step should be t 0.03t mn 0.09(Gρ max ) 1/2. WARNING: these rules are not defntve. Tests wth dfferent parameter values are useful; a skeptcal atttude s advsed!

14 Introducton to Smoothed Partcle Hydrodynamcs Flud equatons n conservaton form: mass: momentum: ρ t + (ρ v) = 0 v t + ( v ) v = Φ + 1 ρ P energy or entropy: u t + ( v )u = P v + u ρ a t + ( v )a = (γ 1)ρ1 γ u Equaton of state (deal gas): P = (γ 1)ρu P = a(s)ρ γ

15 ρ( r), v( r), u( r), a( r) {(m, r, v, u, a ) = 1,...,N} du dt = d r dt = v ( Pρ ) v SPH Formalsm Partcle representaton (c.f. N-body): Densty estmate uses smoothng kernel W( x,h) wth scale h : N ρ( r) m W( r r,h) R where d xw( x,h) = 1 ; estmates of gradents become sums nvolvng gradents of W( x,h). Dynamcal equatons: d v dt = ( Φ) + + u + u vsc da dt ( 1ρ P ) + a vsc = (γ 1)ρ1 γ ( u + u vsc )

16 Comments 1. The smoothng kernel W( x,h) has compact support, so only nearby bodes are ncluded n the sums. Most SPH codes adapt the smoothng length h to the local partcle densty. 2. Adaptve tmesteps are generally necessary to satsfy the Courant condton; most SPH ntegrators are not symplectc. 3. Artfcal vscosty s requred to keep partcles from streamng through shocks. The best formulaton s not entrely clear. 4. SPH s often crtczed as a poor approxmaton to proper gas dynamcs. However, the ISM s much more complex than an deal gas. In the context of galaxy-scale smulatons, momentum and energy conservaton may be all we can expect of a code.