Page - 259 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.4 Oscillating1DSystems:ASecondOrderODE 259 Note that with the choices f(u′) = 0, s(u) = ku, andF(t) = 0 we recover the originalODEu′′+ω2u=0 withω=√k/m. How can we solve (8.66)? As for the simple ODE u′′ +ω2u = 0, we start by rewriting thesecond-orderODEasa systemof two first-orderODEs: v′ = 1 m (F(t)−s(u)−f(v)), (8.68) u′ =v. (8.69) The initial conditionsbecomeu(0)=U0 andv(0)=V0. Any method for a system of first-order ODEs can be used to solve foru(t) and v(t). The Euler-Cromer Scheme An attractive choice from an implementational, ac- curacy, and efficiency point of view is the Euler-Cromer scheme where we take a forwarddifference in (8.68)andabackwarddifference in (8.69): vn+1 −vn Δt = 1 m ( F(tn)−s(un)−f(vn) ) , (8.70) un+1 −un Δt =vn+1, (8.71) We caneasily solve for thenewunknownsvn+1 andun+1: vn+1 =vn+Δt m ( F(tn)−s(un)−f(vn) ) , (8.72) un+1 =un+Δtvn+1 . (8.73) Remarkonthe orderingof the ODEs The ordering of the ODEs in the ODE system is important for the extended model (8.68)–(8.69). Imagine that we write the equation foru′ first and then the one for v′. The Euler-Cromer method would then first use a forward differenceforun+1 andthenabackwarddifferenceforvn+1.Thelatterwould lead toa nonlinearalgebraicequationforvn+1, vn+1 +Δt m f(vn+1)=vn+Δt m ( F(tn+1)−s(un+1) ) , iff(v) is a nonlinear function of v. This would require a numerical method fornonlinearalgebraicequations tofindvn+1,whileupdatingvn+1 througha forwarddifferencegivesanequationforvn+1 that is linearandtrivial tosolve byhand.