Page - 204 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

204 8 SolvingOrdinaryDifferentialEquations is a candidate).1 This is also an example of a first-order ODE, since the highest derivativeappearingin theequationisafirstderivative.Whenthehighestderivative in an ODE is a second derivative, it is a second-order ODE, and so on (a similar terminologyisusedalso forPDEs). OrderofODEversusorderofnumerical solutionmethod Note that an ODE will have an order as just explained. This order, however, shouldnotbeconfusedwith theorderofanumericalsolutionmethodapplied to solve that ODE. The latter refers to the convergencerate of the numerical solutionmethod,addressedat the endof this chapter.We will present several such solution methods in this chapter, being first-order, second-order or fourth-order,forexample(andafirst-orderODEmight, inprinciple,besolved byanyof thesemethods). The present chapter2 starts out preparing for ODEs and the Forward Euler method, which is a first-order method. Then we explain in detail how to solve ODEs numerically with the Forward Euler method, both single (scalar) first-order ODEs and systems of first-order ODEs. After the “warm-up” application—filling of a water tank—aimed at the less mathematically trained reader, we demonstrate all the mathematical and programming details through two specific applications: population growth and spreading of diseases. The first few programs we write, are deliberatelymadeverysimpleandsimilar,whilewe focus thecomputational ideas. Then we turn to oscillating mechanical systems, which arise in a wide range of engineeringsituations.Thedifferentialequationisnowofsecondorder,andtheFor- ward Euler methoddoesnot performtoo well. This observationmotivates the need forothersolutionmethods,andwederivetheEuler-Cromerscheme,thesecond-and fourth-orderRunge-Kutta schemes, as well as a finite difference scheme (the latter tohandle thesecond-orderdifferentialequationdirectlywithout reformulating it as a first-order system). The presentation starts with undamped free oscillations and then treats general oscillatory systems with possibly nonlinear damping, nonlinear spring forces, and arbitrary external excitation. Besides developingprograms from scratch, we also demonstrate how to access ready-made implementations of more advanceddifferentialequationsolvers inPython. As we progress with more advanced methods, we develop more sophisticated and reusable programs. In particular, we incorporategood testing strategies, which allows us to bring solid evidence of correct computations. Consequently, the beginning—with water tank, population growth and disease modeling examples— hasa verygentle learningcurve,while that curvegets significantly steeper towards theendofour sectiononoscillatorysystems. 1 Note that the notation for the derivative may differ. For example, f ′(x) could equally well be writtenas justf ′ (where the dependence on x is to be understood), or as df dx . 2 The reader should be aware of another excellent easy-to-read text by the late Prof. Langtangen, “Finite Difference Computing with Exponential Decay Models” (Springer, open access, 2016, https://www.springer.com/gp/book/9783319294384). Itfits right inwiththematerialonODEsand PDEsof the present book.