19. Class N -body Techniques, Part 2
Time-integration Schemes
Clearly, Newtons laws are IVP. Could use any method (Euler, RK4, etc.). But, issue is to balance accuracy vs. eciency. Typically need many particles to capture dynamics correctly (e.g., in stellar system or galaxy). This consideration may be as important as accuracy of any one individual particle (exception: solar systemN 9, 109 1010 orbits). Could use Euler scheme. But we have seen it is just as easy to design 2nd -order scheme by centering derivatives = could use leapfrog (very stable). Practical timestep control The stability criterion from the discussion of sti systems also applies to the leapfrog integrator for the N-body problem. Can show need t < 2/, where 2 = | F| is a characteristic interaction frequency for a particle (in practice, need t 2/ to avoid problems). But 2 is dierent for every particle; can be very large for particle undergoing close interaction. If have to take max , can be very restrictive. Two solutions: 1. Use dierent timesteps for each particle (individual timesteps). E.g., ti = Fi /Fi eective, but complex implementation, and may break symplecticity of leapfrog (for example). More complex expressions for ti can be formulated, e.g., involving higherorder derivatives of F . These are largely heuristics with convenient properties. It is dicult to prove analytically that one formulation is superior to another. Sometimes ti is discretized, e.g., in factors of 2 (multistepping). 2. Eliminate short-timescale phenomena by modifying gravity on small scales. E.g., Set t = D /30 and/or use softening (see below). Always important to check whether simulation is giving physically meaningful results. Handy technique: reduce timestep by factor of 2 to see if global behavior strongly aected. If so, may have to use smaller steps. Beware of chaos: if state of system strongly dependent on initial conditions, change of timestep may give seemingly vastly dierent results. Need to monitor constants of motion to be sure.
1
Force evaluation
Solving the IVP requires evaluation of the RHS of the ODEs, i.e., must compute interparticle forces. discuss Will PP, PM, P3 M, and tree methods. But rst must consider another practical issue, related to timestep control... Hard interactions Recall F ij = Gmj (ri rj )/|ri rj |3 . Problem: if |ri rj | is small, |F ij | diverges, leading to timestep trouble as |vi | . Physically, very close encounters occur on very short timescales, e.g., can form close binaries with very short periods. To alleviate problem, could use softened forces: F ij = where = softening parameter. Maximum force now Gm2 /2 . Physically, this eliminates possibility of forming binaries with r < . OK when particles represent collection of stars on similar orbits. Not OK if studying small clusters, where each particle represents an individual star. In this case binaries can form and signicantly aect evolution of entire cluster. Modern methods also sometimes use regularization. Binaries (or hierarchies) replaced by pseudo-particles until interaction with other particles becomes important. Gmj (ri rj ) , (|ri rj |2 + 2 )3/2
Direct Summation (PP Method)
Most straightforward way of evaluating Fij .
1 But number of operations = 2 N(N 1) N 2 for N that Fij = Fji .
1 (the
1 2
comes from the fact
10 more particles = 100 more work. Severely limits number of particles that can be...