Seite - 109 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.1 PopulationGrowth 109 Remarkontheworldpopulation Thenumberofpeopleon theplanet2 follows themodelN 0 D r.t/N ,where the net reproduction r.t/ varieswith time and has decreased since its top in 1990. Thecurrentworldvalueof r is 1.2%,and it is difficult topredict futurevalues3. At themoment, thepredictionsof theworldpopulationpoint to agrowth to9.6 billionbeforedeclining. This example shows the limitationof adifferential equationmodel: weneed toknowall input parameters, including r.t/, in order to predict the future. It is seldomthecase thatweknowall inputparameters. Sometimesknowledgeof the solution frommeasurementscanhelpestimatemissing inputparameters. 4.1.7 Verification:ExactLinearSolutionoftheDiscreteEquations Howcanwe verify that the programming of anODEmodel is correct? The best method is tofindaproblemwhere there arenounknownnumerical approximation errors, becausewecan thencompare theexact solutionof theproblemwith the re- sult producedbyour implementationand expect thedifference to bewithin avery small tolerance.Weshall base aunit test on this ideaand implementacorrespond- ing test function (seeSect. 3.4.4) forautomaticverificationofour implementation. It appears thatmostnumericalmethods forODEswill exactly reproduceasolu- tionu that is linear in t.WemaythereforesetuDatCb andchooseanyf whose derivative isa. The choicef.u;t/ D a is very simple, butwemay add anything that is zero, e.g., f.u;t/DaC.u .atCb//m: This is a validf.u;t/ for anya, b, andm. The correspondingODE looks highly non-trivial, however: u0 DaC.u .atCb//m: Using the generalode_FE function in ode_FE.py,wemaywrite a proper test functionas follows (infiletest_ode_FE_exact_linear.py): def test_ode_FE(): """Test that a linear u(t)=a*t+b is exactly reproduced.""" def exact_solution(t): return a*t + b def f(u, t): # ODE return a + (u - exact_solution(t))**m a = 4 b = -1 m = 6 2http://en.wikipedia.org/wiki/Population_growth 3http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/P/Populations.html