Numerically induced stochasticity

On the integration of equations of motion for particle-in-cell codes

“…In particular, area preservation of a non-implicit integration scheme should be expected for a conserved Hamiltonian, i.e., in the absence of damping or amplification in the system. This is not necessarily true for implicit schemes [10]. Further for non-implicit schemes, we shall also see that if J > 1 then the integration scheme is numerically unconditionally unstable and that J = 1 is a necessary condition for numerical stability.…”

“…While the linear stability of time-differenced integration schemes is well-understood as being related to normal modes which are not modes of the exact equation [6][7][8][9], the application of the LF method to nonlinear oscillations was shown by Friedman and Auerbach [10] and Auerbach and Friedman [11] to be limited beyond a certain threshold by ''numerical stochasticity'', which arises from time differencing rather than from any intervening physical process. Auerbach and Friedman [11] also demonstrate the connection between area preservation and long term stability of the computed orbits.…”

“…Auerbach and Friedman [11] also demonstrate the connection between area preservation and long term stability of the computed orbits. Following the Hamiltonian spirit of [10,11], we emphasize here the analogy between a 2nd order time-differencing scheme for an equation of motion and a mapping of phase space (v, z) onto itself. The analogy is useful in as much as the mapping represents a perturbation of an originally integrable Hamiltonian.…”

“…Stochasticity onset occurs around K = 1 [12], i.e., around x B Dt = 1. The isochronous leapfrog method leads to a symmetrized time-centered version of (33) [10] and clearly has the same stochasticity threshold K = 1. The SIMP scheme (13) likewise leads to a standard map.…”

Three dimensional simulations of space charge dominated heavy ion beams with applications to inertial fusion energy

“…The macro-particles are advanced in time using a combination of the "leap frog" and "isochronous leap frog" methods [7]. Each time step goes through the following pattern:…”