Seite - 224 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

224 8 SolvingOrdinaryDifferentialEquations Remarkonthe worldpopulation Thenumberofpeopleontheplanet(http://en.wikipedia.org/wiki/Population_ growth)followsthemodelN′ = r(t)N,wherethenetreproductionr(t)varies with time and has decreased since its top in 1990. The current world value of r is 1.2%, and it is difficult to predict future values. At the moment, the predictions of the world population point to a growth to 9.6 billion before declining. This example shows the limitation of a differential equation model: we need to know all input parameters, including r(t), in order to predict the future. It is seldom the case that we know all input parameters. Sometimes knowledgeofthesolutionfrommeasurementscanhelpestimatemissinginput parameters. 8.2.7 Verification:ExactLinearSolutionoftheDiscrete Equations Howcan we verify that the programmingof an ODE model is correct?One way, is tocomputeconvergencerateswith asolverandconfirmthat the ratesareaccording to expectations. We address convergence rates for ODE solvers later (in Sect. 8.5) andwill thenshowhowacorrespondingtest functionfor theode_FEsolvermaybe written. The best verification method, however, is to find a problem where there are no unknown numerical approximation errors, because we can then compare the exact solutionof the problemwith the resultproducedbyour implementationandexpect the difference to be within a very small tolerance. We shall base a unit test on this idea and implement a corresponding test function (see Sect. 6.6.4) for automatic verificationofour implementation. It appears that most numerical methods for ODEs will exactly reproduce a solution u that is linear in t. We may therefore set u = at + b and choose any f whose derivative is a. The choicef(u,t) = a is very simple, but we may add anythingthat is zero,e.g., f(u,t)=a+(u−(at+b))m. This is a validf(u,t) for any a, b, andm. The corresponding ODE looks highly non-trivial,however: u′ =a+ (u−(at+b))m. Using the generalode_FE function in ode_FE.py, we may write 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."""