Page - 149 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.3 OscillatingOne-DimensionalSystems 149 Fig.4.31 Effect ofnonlinear (left)and linear (right) springonsliding friction U_0 = 0.1 V_0 = 0 T = 2 dt = T/5000. u, v, t = EulerCromer(f=f, s=s, F=F, m=m, T=T, U_0=U_0, V_0=V_0, dt=dt) plot_u(u, t) Running thesliding_frictionfunctiongivesus the results inFig. 4.31with s.u/Dk˛ 1 tanh.˛u/ (left) and the linearizedversion s.u/Dku (right). 4.3.12 AfiniteDifferenceMethod;Undamped,LinearCase Weshall nowaddressnumericalmethods for the second-orderODE u00C!2uD0; u.0/DU0; u0.0/D0; t 2 .0;T ; without rewritingtheODEasasystemoffirst-orderODEs. Theprimarymotivation for“yetanothersolutionmethod”is that thediscretizationprinciplesresult inavery good scheme, andmore importantly, the thinking around the discretization canbe reusedwhensolvingpartial differential equations. The main idea of this numerical method is to approximate the second-order derivativeu00 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: u00.tn/ u nC1 2unCun 1 t2 : (4.74) Theerror in this approximation isproportional to t2. Letting theODEbevalidat somearbitrary timepoint tn, u00.tn/C!2u.tn/D0;