Someone integrates one period of an eccentric Kepler orbit with an embedded…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics
Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness
Problem
Someone integrates one period of an eccentric Kepler orbit with an embedded Runge-Kutta pair using local relative tolerance tol=10−8 on every step, taking about 240 steps per period. They then integrate for 100 periods (about 24000 steps in all) and write: "Since every step met the local tolerance of 10−8, the final position after 100 periods is accurate to a relative error of 10−8." Find the error in this statement, and give an order-of-magnitude estimate of the true relative error after 100 periods, assuming the per-step errors simply add, neither cancelling nor growing.