Seite - 100 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

100 4 SolvingOrdinaryDifferentialEquations Fig.4.1 Mesh in timewithcorrespondingdiscretevalues (unknowns) Fig.4.2 Illustrationof a forwarddifferenceapproximation to thederivative Third, derivatives are to be replaced byfinite differences. To this end,weneed to know specific formulas for howderivatives can be approximated by finite dif- ferences. One simple possibility is to use the definition of the derivative fromany calculusbook, u0.t/D lim t!0 u.tC t/ u.t/ t : Atanarbitrarymeshpoint tn thisdefinitioncanbewrittenas u0.tn/D lim t!0 unC1 un t :