Page - 218 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

218 8 SolvingOrdinaryDifferentialEquations Note thecompactassignment statement numerical_sol = ’bo’ if N_t < 70 else ’b-’ This is a one-linealternative to, e.g., if N_t < 70: numerical_sol = ’bo’ else: numerical_sol = ’b-’ Let us demonstrate a simulation where we start with 100 animals, a net growth rate of 10% (0.1) per time unit, which can be 1 month, and t ∈ [0,20]months. We may first tryΔt of half a month (0.5), which impliesNt = 40. Figure 8.6 shows the results. The solid line is the exact solution, while the circles are the computed numericalsolution.Thediscrepancyisclearlyvisible.What ifwemakeΔt 10times smaller?TheresultisdisplayedinFig.8.7,wherewenowuseasolidlinealsoforthe numericalsolution(otherwise,400circleswouldlookverycluttered,sotheprogram hasa test onhowto display the numerical solution,eitherascirclesora solid line). We can hardly distinguish the exact and the numerical solution. The computing time is also a fraction of a second on a laptop, so it appears that the Forward Euler method is sufficiently accurate for practical purposes. (This is not always true for large,complicatedsimulationmodelsinengineering,somoresophisticatedmethods maybeneeded.) It is also of interest to see what happens if we increase Δt to 2 months. The results inFig. 8.8 indicate that this is an inaccuratecomputation. Fig. 8.6 Evolutionof a population computed with timestep 0.5 month