ASTR415: Problem Set #6
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1 ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal N-body systems wee exploed both quantitatively and qualitatively. Fist, a symmetic two-body poblem was solved using both integatos to compae the esults with the known analytic solution. Then, seveal many-body systems wee geneated and integated with both solves to exploe the effects of softening. In the case of the two-body poblem, the leapfog method did a bette job of conseving enegy than the Runge-Kutta method and esulted in phase diagams that moe closely esembled the exact solution. Fo the many-body systems, howeve, the leapfog method poduced stange esults (such as the paticles aanging themselves into a column) and failed to conseve enegy. Fo N = 5 with the chosen initial conditions, a timestep of h =. was found to be necessay fo the Runge- Kutta method to conseve enegy ove time units ( steps). Finaly, the scaling of the PP method was exploed and found to be less than the theoetical scaling of O(N 2 ). I. INTEGRATION OF THE TWO-BODY PROBLEM The two-body poblem can be educed to a one-body poblem whee a paticle of the educed mass µ = mm2 m +m 2 obits a paticle of the total mass M = m + m 2 and whose position vecto coesponds to the displacement of the two masses fom one anothe, 2. Since the one-body poblem has an analytical solution fo the geomety of the obits, the two-body poblem does as well. This makes it suitable fo testing the N-body code, since the numeical esults fom the integato can be compaed with exact solutions. Conside the case whee the two bodies ae both of unit mass, so m = m 2 =. Choose units such that G =. The obits ae such that the paticles ae sepaated by unit distance at apoapsis. Choosing this as the initial configuation, at t =, = /2,, and 2 = /2,,. The initial velocities detemine the eccenticity e of the obit though the elations a = 2 v2 GM a = ( + e)a whee a = is the paticles sepaation at apoapsis. Once v is chosen, the initial velocity conditions become v 2 =, v/2, and v 2 =, v/2,. Fo e =.5, v =. The obits of both paticles in such a system ae plotted in figue, as integated using both the leapfog and Runge-Kutta methods. As the peiod of these obits is 2.4 time units, each obit equies 48 time steps using a step size of h =.5. Theefoe, about 5 steps ae equied to achieve complete obits. Repoting the esults of the integation evey 5 steps yields points with which to analyze the obit ove time. Fom this data, the elative adial velocity v of the paticles at each point in time can be computed and plotted against thei sepaation distance, poducing the phase diagam shown in figue 2. Futhemoe, the enegy can be calculated accoding to E = 2 v2 + 2 v2 2 The enegy of the system as calculated by both integatos is plotted against time in figue 3. Looking at both of these plots, it is clea that the Leapfog method does a much bette job at conseving enegy than the Runge-Kutta method. Howeve, the Leapfog method is vey poo at maintaining the positions of the paticles, and they tend to pecess apidly, unlike the obits poduced by the Runge-Kutta method. Fo e =.9, v = /5. The obits of the paticles in this system ae plotted in figue 4 fo both integatos. With an obital peiod of.7 time units, each obit equies 565 time steps using a step size of h =.3. Theefoe, appoximately 57 steps ae equied to achieve complete obits, and an output fequency of once evey 57 steps yields data points along these obits. The phase diagam fo this system is shown in figue 5. Neithe integato can accuately epoduce the exact solution nea peiapsis, but the leapfog method does a fa bette job at apoapsis than the Runge-Kutta method. The enegy of the system is plotted against time in figue 6. Hee the supeio ability of the leapfog method to conseve enegy ove the Runge-Kutta method is even moe ponounced.
2 2 Obits fom Leapfog Method (e=.5) Obits fom Runge-Kutta Method (e=.5) y y x x FIG. : Plots of the obits fo e =.5. The obit of one paticle is plotted with lines while the obit of the othe is plotted with cosses. Phase Plot fo Leapfog Method (e=.5) Phase Plot fo Runge-Kutta Method (e=.5) Phase Plot fo Exact Solution (e=.5) v v v FIG. 2: Phase diagam fo e =.5. A. Solving Keple s Equation fo Radial Velocity In figues 2 and 5, a phase diagam fo the exact solution is given. Keple s equation has no analytic solution expessing as a function of t, but it does have an analytic solution fo ṙ in tems of. This solution can be found as follows. Keple s equation states that M = E e sin(e) whee M is the mean anomaly, defined as M = 2πt/P, and E is the eccentic anomaly, defined as cos ( /a e ). Diffeentiating both sides with espect to t and solving fo ṙ yields When e =.5 this simplifies to ṙ = 2( )(e + + e )
3 Enegy Consevation (e=.5) Leapfog Runge-Kutta Enegy (E) Time (t) FIG. 3: Enegy vesus time fo e =.5.. Obits fom Leapfog Method (e=.9) Obits fom Runge-Kutta Method (e=.9) y y x x FIG. 4: Plots of the obits fo fo e =.9. The obit of one paticle is plotted with lines while the obit of the othe is plotted with cosses. and when e =.9, it simplifies to These equations wee used to make the phase diagams fo the exact solutions in ode to compae them with those poduced by the integated solutions.
4 4 Phase Plot fo Leapfog Method (e=.9) Phase Plot fo Runge-Kutta Method (e=.9) Phase Plot fo Exact Solution (e=.9) v v v FIG. 5: Phase diagam fo e = Enegy Consevation (e=.9) Leapfog Runge-Kutta Enegy (E) Time (t) FIG. 6: Enegy vesus time fo e =.9. II. THE N-BODY PROBLEM A. Initial Conditions Seveal simulations wee done using N = 5 and N = points distibuted in a sphee with adius. The masses wee distibuted accoding to a Rayleigh distibution centeed at eithe o 2. The system was given a positive angula momentum along the z-axis by choosing initial velocities in the x y plane as if each paticle wee in a cicula obit about a paticle of unit mass at the oigin. The z velocities wee chosen unifomly as small deviations fom. Fo the fist two, low-pecision, uns (N = ), which detemined the effects of softening, the mass distibution was centeed at mass unit and the initial velocities wee not scaled. The softening paamete was fist set to and then to.2, and esults wee obtained fo both integatos. Fo subsequent, high-pecision, uns (N = 5), the mass distibution was centeed at 2 mass units, and the initial velocities wee scaled by a facto of. The initial conditions used in the latte case ae shown in figue 7.
5 5 FIG. 7: Initial conditions used fo the high-pecision simulations (N = 5). On the left ae the initial positions of the bodies with the glyphs coloed and scaled accoding to mass. On the ight ae the initial velocities. B. Qualitative Results Using the Runge-Kutta method with a timestep of h =. and no softening, paticles quickly begin to leave the system at high velocities due to close encountes, and the esults of the simulation ae clealy non-physical. Adding a softening paamete of.2 geatly educes this effect, and the system quickly collapses in on itself befoe ebounding and sending some paticles into a loose extenal shell while the othes fomed a dense inteio coe with a elatively high chaacteistic velocity. Integating a smalle system with a smalle timestep poduced simila esults, except that, as a esult of the geate voticity in the initial conditions, the paticles fist fomed an annulus befoe collapsing in on themselves. In both cases the leapfog method appeaed to give unphysical esults, with the paticles foming a loose column. Howeve, with the smalle timestep, both methods give simila esults fo ealy times. C. Enegy Consevation The total enegy of the system should be given by N E = 2 m ivi 2 i= This epesents the sum of the kinetic enegy of each of the paticles and the potential each paticle contibutes to those counted afte it. Supisingly, the leapfog integato was unable to conseve the enegy of the system ove time units even fo time steps as small as h =. (see figue 8). Fo lage time steps, the Runge-Kutta integato exhibited even lage deviations in enegy, but as the time step was educed, the enegy emained stable until nea the end of the simulation (see figue 9). Given the leapfog method s ability to conseve enegy fo smalle systems and the chaacteistic shape of its enegy cuves in all fou simulations, it is possible that an eo exists in eithe the enegy computation o simulation code. Howeve, the Runge-Kutta method seems to hold the enegy steady fo small time steps. Fo these initial conditions and on this time scale, a time step of h =. appeas to be necessay to achieve meaningful esults. Unfotunately, even fo as few as N = 5 points, this takes seveal hous to compute on a moden pocesso. N j=i+ m i m j ij D. Pogam Scalability The PP method fo solving the N-body poblem is an O(N 2 ) algoithm. Howeve, the code appeas to scale at an ode smalle than 2. Random initial conditions wee geneated fo N between 64 and 24 in powes of 2, and the
6 6 Enegy fom Leapfog Method (h=., no softening) Enegy fom Leapfog Method (h=., softening) Enegy - Enegy Time [t*] Time [t*] Enegy fom Leapfog Method (h=.) Enegy fom Leapfog Method (h=.) Enegy -3 Enegy Time [t*] Time [t*] FIG. 8: Consevation of enegy by the leapfog integato fo modeate N. time needed to integate the system fo steps was ecoded fo both the leapfog method and the Runge-Kutta method on two diffeent computes. The esults ae plotted in figue. On both machines, the leapfog method scaled slightly bette than the Runge-Kutta method. On the AMD Athlon64, the leapfog method scaled as O(N.6 ), while the Runge-Kutta method scaled as O(N.9 ). On the Intel Coe2 Duo, the leapfog method scaled as O(N.3 ), while the Runge-Kutta method scaled as O(N.6 ). Fo both methods the Intel Coe2 Duo pefomed significantly bette (by a facto of 2) despite the code being single-theaded. These scaling exponents ae difficult to explain, as the nested sum pesent in evey function evaluation is clealy an O(N 2 ) opeation. Futhemoe, as the application is witten in Java, SIMD instuctions ae not being used to acceleate the loops. In some situations it is possible fo ovehead to ceate the illusion of a smalle scaling coefficient, but that is not the case hee, as the slopes of the cuves ae steady fo lage N. If the poblem wee to gow beyond the cache size of the pocesso, one would expect the oveall slope to be atificially geate than the tue value, not less. The low scaling exponents fo this pogam ae at pesent a mystey.
7 7 Enegy fom Runge-Kutta Method (h=., no softening) Enegy fom Runge-Kutta Method (h=., softening).2e+7 e+7 -e+7 8e+6-2e+7 6e+6 Enegy 4e+6 Enegy -3e+7 2e+6-4e+7-5e+7-2e e Time [t*] Time [t*] Enegy fom Runge-Kutta Method (h=.) Enegy fom Runge-Kutta Method (h=.) Enegy -78 Enegy Time [t*] Time [t*] FIG. 9: Consevation of enegy by the Runge-Kutta integato fo modeate N.
8 8 N-Body Scaling Leapfog on AMD Athlon64 Leapfog on Intel Coe2 Duo Runge-Kutta on AMD Athlon64 Runge-Kutta on Intel Coe2 Duo Execution Time. Numbe of Bodies (N) FIG. : Execution time vesus N.
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