问题
I was under the impression that the algorithm should conserve energy if the system being modelled does. I'm modelling the solar system, which should conserve energy. The program conserves angular momentum and does produce stable orbits, but the total energy (kinetic + gravitational potential) oscillates around some baseline. The oscillations are significant. Are there common reasons why this might happen?
Model assumes planets are point masses, circular orbits (I've also tried elliptical orbits and the energy still oscillates) and uses Newtonian mechanics. I can't think what other features of the program might be affecting the outcome.
If it is just expected that the energy oscillates, what causes that??
回答1:
Merged from the comments: For a full gravitational N-body problem, I don't think any numerical integrator will be symplectic. Velocity Verlet isn't symplectic even for a single point orbiting a center (easy to check, since it has a trivial analytical solution with g = v^2/R). So I suggest trying a higher-order integrator (such as Runge-Kutta), and if energy deviations almost go away (meaning the the calculations are generally correct), you can re-scale the combined kinetic energy to keep the total energy conserved explicitly. Specifically, you compute the updated Ekin_actual and Ekin_desired = Etotal_initial - Epotential, and scale all velocities by sqrt(Ekin_desired / Ekin_actual)
回答2:
Look up the Verlet-Störmer paper by Hairer et al. (Geometric numerical integration illustrated by the Störmer/Verlet method). There should be several sources online.
In short, a symplectic integrator preserves a Hamiltonian and thus energy, but it is a modified Hamiltonian. If the method is correctly initialized, the modification is a perturbation of order O(h²), where h is the step size. Incorrect initialization gives a perturbation of O(h), while the observed oscillation should still have an amplitude of O(h²).
Thus the observed pattern of an oscillating energy as computed by the physical formulas is completely normal and expected. An error would be observed if the energy were to (rapidly) deviate from this relatively stable pattern.
An easy, but slightly inefficient method to get from the order 2 Verlet method an order 4 symplectic integrator is to replace
Verlet(h)
by
Verlet4(h) {
Verlet(b0*h);
Verlet(-b1*h);
Verlet(b0*h);
}
where b0=1/(2-2^(1/3))=1.35120719196… and b1=2*b0-1=1.70241438392…. This is called a "composition method".
来源：https://stackoverflow.com/questions/34651818/velocity-verlet-algorithm-not-conserving-energy