Page - 137 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.3 OscillatingOne-DimensionalSystems 137 Fig.4.22 Illustrationof the impact of resolution (timestepsperperiod) and lengthof simulation right). However, after 194–200periods the errors havegrown (lower left), but can besufficiently reducedbyhalving the timestep (lower right). With all themethods inOdespy at hand, it is noweasy to start exploring other methods, such as backward differences instead of the forward differences used in theForwardEuler scheme. Exercise4.17addresses thatproblem. Odespy contains quite sophisticated adaptivemethodswhere the user is “guar- anteed” toget a solutionwith prescribedaccuracy. There is nomathematical guar- antee, but the error will for most cases not deviate significantly from the user’s tolerance that reflects the accuracy. A very popular method of this type is the Runge-Kutta-Fehlbergmethod, which runs a 4th-order Runge-Kuttamethod and uses a 5th-orderRunge-Kuttamethod to estimate the error so that t can be ad- justed to keep the error below a tolerance. Thismethod is alsowidely known as ode45, because that is the nameof the function implementing themethod inMat- lab. Wecan easily test theRunge-Kutta-Fehlbergmethod as soon asweknow the correspondingOdespyname,which isRKFehlberg: compare(odespy_methods=[odespy.EulerCromer, odespy.RKFehlberg], omega=2, X_0=2, number_of_periods=200, time_intervals_per_period=40)