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
The same circular orbit at a=1AU, integrated at Δt=10−3yr for 100 years, gives these fractional energy drifts ΔE/∣E0∣: Euler +0.66 (still rising), Euler-Cromer bounded within 3.9×10−5, Verlet bounded within 3.9×10−10, and Runge-Kutta −1.7×10−10 (a loss, still falling steadily). A bounded method's error stays within the same fixed band forever, however long the run; a method whose error is a steady loss accumulates that loss in rough proportion to the elapsed time.
((a)) Predict the size of the Runge-Kutta drift after 104 years (a hundred times longer), assuming the loss continues to accumulate in proportion to elapsed time.
((b)) Give the new ranking of the four methods by ΔE/∣E0∣ at 104 years, smallest first, and say which pair of methods has swapped places compared with the 100-year ranking (Runge-Kutta smallest, then Verlet, then Euler-Cromer, then Euler).
((c)) State in one sentence why the 100-year ranking is not a safe guide to which method is best for a much longer run.