Seite - 101 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.1 PopulationGrowth 101 Instead of going to the limit t ! 0wecanuse a small t, which yields a com- putableapproximation tou0.tn/: u0.tn/ u nC1 un t : This isknownasa forwarddifference sincewegoforward in time(unC1) tocollect informationinu toestimate thederivative. Figure4.2 illustrates the idea. Theerror of the forwarddifference isproportional to t (oftenwrittenasO. t/, butwewill notuse thisnotation in thepresentbook). Wecannowplug in the forwarddifference in ourdifferential equation sampled at thearbitrarymeshpoint tn: unC1 un t Df.un;tn/; (4.3) orwithf.u;t/D ru inour specialmodelproblemforpopulationgrowth, unC1 un t D run: (4.4) If r dependson time,we insert r.tn/D rn forr in this latter equation. The fourth step is to derive a computational algorithm. Looking at (4.3), we realize that ifun shouldbeknown,wecan easily solvewith respect tounC1 to get a formula foruat thenext time level tnC1: unC1 DunC tf.un;tn/: (4.5) Providedwehaveaknownstartingvalue,u0 DU0,wecanuse(4.5) toadvancethe solutionbyfirst computingu1 fromu0, thenu2 fromu1,u3 fromu2, and so forth. Suchanalgorithmiscalledanumerical scheme for thedifferential equationand oftenwrittencompactlyas unC1 DunC tf.un;tn/; u0 DU0; nD0;1;:: :;Nt 1: (4.6) This scheme isknownas theForward Euler scheme, alsocalledEuler’smethod. Inour specialpopulationgrowthmodel,wehave unC1 DunC trun; u0 DU0; nD0;1;:: :;Nt 1: (4.7) Wemayalsowrite thismodel using theproblem-specific symbolN insteadof the genericu function: NnC1 DNnC trNn; N0 DN0; nD0;1;:: :;Nt 1: (4.8) The observant reader will realize that (4.8) is nothing but the computational model (4.2) arising directly in the model derivation. The formula (4.8) arises, however, fromadetour via a differential equation and anumericalmethod for the differential equation. This looks rather unnecessary! The reasonwhywebother to