Page - 279 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.6 Exercises 279 Exercise8.15:The LeapfrogMethod We consider the general ODE problemu′(t)= f(u,t),u(0)=U0. To solve such anODEnumerically,thesecondorderLeapfrogmethodapproximatesthederivative (at some time tn=nΔt) byuseofacentereddifferenceover two time steps, u′(tn)≈ u(tn+1)−u(tn−1) 2Δt = u n+1 −un−1 2Δt . a) Replace the derivative in the ODE by the given centered difference approxima- tionandshowthat this allowsus to formulate: un+1 =un−1+2Δtf(un,tn), n=1,2,.. .,Nt −1, withu0 =U0.Dowe have the informationwe need toget theschemestarted? b) Theproblemyoudiscovered in thepreviousquestion,maybefixedbyusing the ForwardEulermethod.However, theLeapfrogmethodisasecondordermethod, while the ForwardEulermethod isfirst order. Argue, with reference to the Taylor series (see, e.g., Exercise 8.14), why the Forward Euler method can be used without reducing the order of the overall scheme. c) ImplementtheLeapfrogschemeinafunctionleapfrog.Makesurethefunction takesanappropriateset of inputparameters, so that it is easy to importanduse. d) Write a function compare_FE_leapfrog that compares graphically the solu- tionsproducedbytheForwardEulerandLeapfrogmethods,whentheysolve the population growth modelu′ = 0.1u, withu(0) = 100. Let the total time span T =20,anduse a time stepdt=2. In the plotproduced, includealso theexact solution, so that thenumerical solutionscanbeassessed. e) Suggest a reasonable asymptotic error model before you write a proper test function test_convergence_rates that may be used to compute and check the convergence rates of the implemented Leapfrog method. However, the test functionshould takeappropriate inputparameters, so that it canbeusedalso for otherODE solvers, in particular theode_FE implementedpreviously. Include your test function in a program, together with the two functions you defined previously (leapfrog and compare_FE_leapfrog). Write the code with a test block, so that it getseasy toeither import functionsfromthe module, or to run it as aprogram. Finally, run theprogram(so thatcompare_FE_leapfroggetscalled,aswell as test_convergence_rates for both FE and Leapfrog) and confirm that it works as expected. In particular, does the plot look good, and do you get the convergenceratesyouexpectedforForwardEulerandLeapfrog? Filename:growth_leapfrog.py.