Page - 128 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

128 4 SolvingOrdinaryDifferentialEquations getu1 D2,v1 D 1:25663706,u2 D1:80260791, andv2 D 2:51327412. Such calculations show that the program is seemingly correct. (Later, we can use such values toconstruct aunit test andacorrespondingtest function.) Thenext step is to reduce the discretization parameter t and see if the results becomemore accurate. Figure 4.17 shows the numerical and exact solution for the cases t D P=40;P=160;P=2000. The results clearly become better, and the finest resolution gives graphs that cannot be visually distinguished. Neverthe- less, the finest resolution involves 6000 computational intervals in total, which is considered quitemuch. This is no problemon amodern laptop, however, as the computations take just a fractionofa second. Although 2000 intervals per oscillation period seem sufficient for an accurate numerical solution, the lower rightgraph inFig. 4.17 shows that ifwe increase the simulation time, here to20periods, there is a little growthof the amplitude,which becomes significant over time. The conclusion is that the Forward Eulermethod has a fundamental problemwith its growing amplitudes, and that a very small t is required to achieve satisfactory results. The longer the simulation is, the smaller t has tobe. It is certainly time to lookformoreeffectivenumericalmethods! Fig.4.17 Simulation of an oscillating systemwith different time steps. Upper left: 40 steps per oscillation period. Upper right: 160 steps per period. Lower left: 2000 steps per period. Lower right: 2000 stepsperperiod, but longer simulation