Page - 253 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.4 Oscillating1DSystems:ASecondOrderODE 253 Fig. 8.25 Illustration of the impactof resolution (timesteps per period) and length of simulation easily test theRunge-Kutta-Fehlbergmethodassoonasweknowthecorresponding Odespyname,which isRKFehlberg: compare(odespy_methods=[odespy.EulerCromer, odespy.RKFehlberg], omega=2, X_0=2, number_of_periods=200, time_intervals_per_period=40) Note that the time_intervals_per_period argument refers to the time points where we want the solution. These points are also the ones used for numerical computationsintheodespy.EulerCromersolver,while theodespy.RKFehlberg solver will use an unknown set of time points since the time intervals are adjusted as themethodruns.Onecaneasily lookat thepointsactuallyusedbythemethodas theseareavailableasanarraysolver.t_all(butplottingorexaminingthepoints requiresmodifications inside thecomparemethod). Figure 8.26 shows a computational example where the Runge-Kutta-Fehlberg methodisclearlysuperior to theEuler-Cromerschemein longtimesimulations,but thecomparison is not really fair because the Runge-Kutta-Fehlbergmethodapplies about twice as many time steps in this computationand performsmuch more work per time step. It is quite a complicated task to compare two so different methods in a fair way so that the computational work versus accuracy is scientifically well reported.