Page - 247 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.4 Oscillating1DSystems:ASecondOrderODE 247 Fig. 8.23 Illustration of a centered difference approximation to the derivative first thinkingofa scalarODE, is to forma centereddifferenceapproximationto the derivativebetween two timepoints: u′(tn+ 1 2 Δt)≈ u n+1 −un Δt . Thecentereddifferenceformula isvisualized inFig.8.23.Theerror in thecentered difference is proportional toΔt2, one order higher than the forward and backward differences,which means that if we halveΔt, the error is more effectively reduced in the centered difference since it is reduced by a factor of four rather than two. The problem with such a centered scheme for the general ODEu′ = f(u,t) is thatweget un+1−un Δt =f(un+12,t n+12), which leads to difficulties since we do not know whatun+12 is. However, we can approximatethevalueoff betweentwotimelevelsbythearithmeticaverageof the valuesat tn and tn+1: f(un+ 1 2,t n+12)≈ 1 2 (f(un,tn)+f(un+1,tn+1)). This results in un+1 −un Δt = 1 2 (f(un,tn)+f(un+1,tn+1)), which in general is a nonlinear algebraic equation for un+1 if f(u,t) is not a linear functionofu.Todealwith theunknowntermf(un+1,tn+1),withoutsolving