Page - 215 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.2 PopulationGrowth:AFirstOrderODE 215 is constant.Thisproperty implies that tn=nΔt, n=0,1,.. .,Nt . Second, the differential equation is supposed to hold at the mesh points. Note that this is an approximation,because the differential equation is originallyvalid for all realvaluesof t. We canexpress thispropertymathematicallyas u′(tn)=f(un,tn), n=0,1,.. .,Nt . Forexample,withourmodelequationu′ = ru,we have thespecial case u′(tn)= run, n=0,1,.. .,Nt, or u′(tn)= r(tn)un, n=0,1,.. .,Nt, if r dependsexplicitlyon t. Third, derivatives are to be replaced by finite differences. To this end, we need to know specific formulas for how derivatives can be approximated by finite differences.Onesimplepossibility is touse thedefinitionof thederivativefromany calculusbook, u′(t)= lim Δt→0 u(t+Δt)−u(t) Δt . Atanarbitrarymeshpoint tn thisdefinitioncanbewrittenas u′(tn)= lim Δt→0 un+1 −un Δt . Instead of going to the limit Δt → 0 we can use a small Δt, which yields a computableapproximationtou′(tn): u′(tn)≈ u n+1 −un Δt . This isknownasa forwarddifferencesincewe goforward in time(un+1) tocollect information inu to estimate the derivative.Figure 8.4 illustrates the idea.The error of the forwarddifference isproportional toΔt (oftenwrittenasO(Δt), butwewill notuse thisnotation in thepresentbook). We can now plug in the forward difference in our differential equation sampled at thearbitrarymeshpoint tn: un+1−un Δt =f(un,tn), (8.3)