Parallel KS Block-Step Method. Sverre Aarseth. Institute of Astronomy, Cambridge
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1 Parallel KS Block-Step Method Sverre Aarseth Institute of Astronomy, Cambridge Code Overview Hermite KS Prediction & Correction Iteration Time-Steps Decision-Making Binary Project
2 Code Overview Directories Ncode, Docs, Chain, Nchain, ARchain, ARint, Block Standard make nbody6 in dir Ncode nbody6 Block-steps make nbody6 in dir Block nbody6b NBODY7 make -f Makefile7 nbody6 in Ncode nbody7 NBODY7b make -f Makefile7 nbody6 in Block nbody7b GPU make gpu in dir GPU2 produces nbody6.gpu GPUb make -f Makefile6 gpu nbody6b.gpu ARC make -f Makefile7 gpu nbody7.gpu ARCb make -f Makefile8 gpu nbody7b.gpu
3 Program Flow KS predict Predict u, u to order u (5) Perturbers Predict r, ṙ and obtain F, Ḟ Corrector Temporary correct u,u & h Iteration One or more, same F, Ḟ Final stage Form u (4), u (5), copy to common Time Derivatives up to t (5) or t (6) Time-step Set τ, convert to t & truncate Invert Newton Raphson gives τ bs
4 Hermite KS Standard KS New notation u h t = 2 hu + 2 R LT F kl = 2u L T F kl = u u P = F kl F u = u Q = L T P, Basic equations F u = 2 hu + 2 RQ h t = 2u Q = u u Hermite F, F formulation F u = 2 hu + 2 RQ F u = 2 (h u + hu + R Q + RQ ) = 2u Q h = 2F u Q + 2u Q t = u u h
5 Hermite KS with block-steps Predict u, u, h to highest order u pred = u + τu + τ2 2 u + τ3 6 u + τ4 24 u 4 + τ5 20 u u pred = u + τu + τ2 2 u + τ3 6 u 4 + τ4 24 u 5, h pred = h + τh + τ 2 h + τ 6 h + τ 24 h. Force evaluation P = L T uf, P = L T u F + rl T u F. (2) Derivatives for corrector u = (hu + rp), 2 u = 2 (h u + hu + r P + rp ), h = 2(u P), h = 2(u P + u P ). (3)
6 Fourth-order corrector (Hut, Makino & McMillan 995) u corr = u old + τ 2 (u corr + u old ) τ2 2 (u new u old ), u corr = u old + τ 2 (u new + u old ) τ2 2 (u new u old ), h corr = h old + τ 2 (h new + h old ) τ2 2 (h new h old ). (4) Velocity derivatives for prediction u 4 mid = u new u τ u 5 mid = 2 τ 2 old, ( u new + u old 2 u new u ) old. (5) τ Shifted from midpoint u 4 new = u 4 mid + τ 2 u 5 mid, u 5 new = u 5 mid, h new = h mid + τ h new = h 2 h mid, mid. (6)
7 Taylor series for physical time t = u u, t = 2(u u ), t = 2(u u + u u ), t 4 = 2(u u ) + 6(u u ), t 5 = 2(u u 4 + 8(u u ) + 6(u u ), t 6 = 2(u u 5 + 0(u u (u u ). (7) Next look-up time t next = t + 6 n= t n τ n. (8) n! Regularized time-step by inversion t + 6 n= t n τ n n! = 0. (9) Newton-Raphson iteration to solve for τ x i+ = x i f(x i) f (x i ).
8 Initial guess τ = t bs t τ orig, with quantized time-step t bs (= 2 n t orig ). Commensurability condition mod (t sys, t bs ) = 0. Solve equation (9) to obtain τ bs.
9 0.000 ECC = 0.96 e-05 DT e-06 e e Time
10 0.03 ECC = DTU e Time
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13 Massive KS Binary Initial condition e, a, m, m 2, c GR coalescence R coal = 6(m + m 2 ) c 2 PN perturbation PN & 2.5PN Isolated binary e = 0.99, t/t K = 0 5 Time-step τ = η u 2 h /2, N = 2π η u η u = 0., it = 3, E/E = η u = 0.3, it = 3, E/E = 0 8 η u = 0.3, it = 4, E/E = 3 0
14 NBODY7 Binary Project Initial condition e, a, m, m 2, c Perturbation PN & 2.5PN in pnpert.f Energy loss Perturbed KS or Peters 964 Accumulated Hermite, previous Ė GR, Ë GR Integration E GR = 2 (Ė 0 + Ė) t + 2 (Ë 0 Ë) t 2 Identity check h < 4 h
15 PN PN+2.5PN Peters SEMI e-05 e-06 e Time
16 Conclusions Performance Significant speed-up Accuracy One or two extra iterations Memory No new variables needed Time-steps Similar to current method Usage Special procedures require attention
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