Page - 266 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

266 8 SolvingOrdinaryDifferentialEquations Fig. 8.34 Effect of nonlinear (left) and linear (right) spring on sliding friction 8.4.12 AFiniteDifferenceMethod;Undamped,LinearCase We shallnowaddressnumericalmethodsfor thesecond-orderODE u′′+ω2u=0, u(0)=U0, u′(0)=0, t∈ (0,T], without rewriting theODEasasystemoffirst-orderODEs.Theprimarymotivation for“yetanothersolutionmethod”is that thediscretizationprinciplesresult inavery good scheme, and more importantly, the thinking around the discretization can be reusedwhensolvingpartialdifferentialequations. The main idea of this numerical method is to approximate the second-order derivative u′′ by a finite difference. While there are several choices of difference approximations to first-order derivatives, there is one dominating formula for the second-orderderivative: u′′(tn)≈ u n+1−2un+un−1 Δt2 . (8.74) The error in this approximation is proportional toΔt2. Letting the ODE be valid at somearbitrary time point tn, u′′(tn)+ω2u(tn)=0, wejust insert theapproximation(8.74) toget un+1 −2un+un−1 Δt2 =−ω2un. (8.75) We nowassumethatun−1 andun arealreadycomputedand thatun+1 is thenew unknown.Solvingwith respect toun+1 gives un+1 =2un−un−1 −Δt2ω2un. (8.76)