Recall the stability argument for a restoring force: Euler's method is…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Simulating classical dynamical systems
Computational Physics, Ch An orbit that must not drift
Problem
Recall the stability argument for a restoring force: Euler's method is unconditionally unstable and its solution spirals away from the truth; Euler-Cromer and velocity Verlet are stable and their energy error stays inside a fixed band forever; fourth-order Runge-Kutta has the smallest truncation error per step of the four but is not built to conserve any modified energy, so its energy error accumulates steadily rather than staying bounded. A circular orbit at a=1AU is integrated for 100 years at Δt=10−3yr by each of the four methods. Assume that over a run this short the steady Runge-Kutta loss has not yet had time to overtake either bounded band, so that at 100 years each method's drift is still set by the size of its error per step.
((a)) Using only the argument above, predict the order of the four methods by the magnitude of their fractional energy drift ΔE/∣E0∣ at the end of the 100 years, smallest first.
((b)) For each method, state whether its drift is bounded or grows with elapsed time.