Page - 282 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

282 8 SolvingOrdinaryDifferentialEquations Exercise8.18:The Three-StepAdams-BashforthMethod ThisexercisebuildsonExercise8.17,soyoubetterdothatonefirst.Anothermulti- stepmethod,is the three-stepAdams-Bashforthmethod.This isa thirdordermethod witha computationalscheme that reads: un+1 =un+Δt 12 ( 23f(un,tn)−16f(un−1,tn−1)+5f(un−2,tn−2) ) . forn=2,3,.. .,Nt −1,withu0 =U0. a) Assumethat someoneimplemented theschemeas follows: def adams_bashforth_3(f, U_0, dt, T): """Third-order Adams-Bashforth scheme for solving first order ODE""" N_t = int(round(T/dt)) u = np.zeros(N_t+1) t = np.linspace(0, N_t*dt, len(u)) u[0] = U_0 # Compute missing starting values u[1] = 100*np.exp(0.1*dt) u[2] = 100*np.exp(0.1*(2*dt)) for n in range(1, N_t, 1): u[n+1] = u[n] + (dt/12)*(23*f(u[n], t[n]) - \ 16*f(u[n-1], t[n-1]) +\ 5*f(u[n-2], t[n-2])) return u, t There is one (known!)bug here,find it! Try first by simply reading the code. If not successful,youmaytry to run it anddosometesting onyourcomputer. Also, what would you say about the way that missing starting values are computed? b) RepeatExercise8.17,using thegiven three-stepmethod in steadof the two-step method. Note that with the three-step method, you need 3 starting values. Use the Runge-Kutta third order scheme for this purpose. However, check also the convergencerate of theschemewhenmissingstartingvaluesarecomputedwith ForwardEuler in stead. Filename:Adams_Bashforth_3.py. Exercise8.19:Use aBackwardEuler SchemeforOscillations Consider(8.43)–(8.44)modelinganoscillatingengineeringsystem.This2×2ODE systemcanbesolvedbytheBackwardEulerscheme,which isbasedondiscretizing derivatives by collecting information backward in time. More specifically,u′(t) is approximatedas u′(t)≈ u(t)−u(t−Δt) Δt .