Page - 283 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.6 Exercises 283 A general vector ODE u′ = f(u,t), where u and f are vectors, can use this approximationas follows: un−un−1 Δt =f(un,tn), which leads toanequationfor the newvalueun: un−Δtf(un,tn)=un−1 . Fora generalf , this is a systemofnonlinearalgebraicequations. However, theODE(8.43)–(8.44)is linear, so aBackwardEuler scheme leads to asystem of two algebraicequationsfor twounknowns: un−Δtvn=un−1, (8.89) vn+Δtω2un=vn−1 . (8.90) a) Solve thesystemforun andvn. b) Implement the found formulas for un and vn in a program for computing the entirenumerical solutionof (8.43)–(8.44). c) Run the program with a Δt corresponding to 20 time steps per period of the oscillations (see Sect. 8.4.3 for how to find such aΔt). What do you observe? Increase to 2000 time steps per period. How much does this improve the solution? Filename:osc_BE.py. Remarks While the Forward Euler method applied to oscillation problemsu′′ + ω2u = 0 gives growing amplitudes, the Backward Euler method leads to signifi- cantlydampedamplitudes. Exercise8.20:Use Heun’sMethod for theSIR Model Makea programthat computes the solutionof the SIRmodel fromSect. 8.3.1both by the Forward Euler method and by Heun’s method (or equivalently: the second- order Runge-Kutta method) from Sect. 8.4.5. Compare the two methods in the simulation case from Sect. 8.3.3. Make two comparison plots, one for a large and one for a small time step. Experiment to find what “large” and “small” should be: the large one gives significant differences, while the small one lead to very similar curves. Filename:SIR_Heun.py. Exercise8.21:Use OdespytoSolvea Simple ODE Solve u′ =−au+b, u(0)=U0, t∈ (0,T]