Page - 244 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

244 8 SolvingOrdinaryDifferentialEquations Fig. 8.20 Simulation of an oscillating system with different time steps. Upper left: 40 steps per oscillation period. Upper right: 160 steps per period. Lower left: 2000 steps per period. Lower right:2000 steps perperiod, but longer simulation wecanreplaceun in the lastequationbytherecentlycomputedvalueun+1 fromthe first equation: un+1 =un+Δtvn, (8.49) vn+1 =vn−Δtω2un+1 . (8.50) Before justifying this fix more mathematically, let us try it on the previous example.The results appear in Fig. 8.21.We see that the amplitudedoes not grow, but the phase is not entirely correct. After 40 periods (Fig. 8.21 right) we see a significantdifferencebetween the numerical and the exact solution. DecreasingΔt decreasestheerror.Forexample,with2000intervalsperperiod,weonlyseeasmall phaseerrorevenafter50,000periods(!).We cansafelyconcludethat thefixresults inanexcellentnumericalmethod! Letus interpret theadjustedschememathematically.Firstweorder(8.49)–(8.50) such that thedifferenceapproximationsto derivativesbecometransparent: un+1 −un Δt =vn, (8.51) vn+1 −vn Δt =−ω2un+1 . (8.52)