Page - 139 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.3 OscillatingOne-DimensionalSystems 139 Fig.4.24 The last10of40periodsofoscillationsby the4th-orderRunge-Kuttamethod Thealgorithm Wefirst just state the four-stagealgorithm: unC1 DunC t 6 fnC2 OfnC12 C2 QfnC12 C NfnC1 ; (4.59) where OfnC12 Df unC 1 2 tfn;tnC12 ; (4.60) QfnC12 Df unC 1 2 t OfnC12 ;tnC12 ; (4.61) NfnC1 Df unC t QfnC12 ;tnC1 : (4.62) Application WecanrunthesamesimulationasinFigs.4.16,4.18,and4.21,for40 periods. The10 last periodsare shown inFig. 4.24. The results lookas impressive as thoseof theEuler-Cromermethod. Implementation The stages in the 4th-order Runge-Kuttamethod can easily be implemented as amodification of theosc_Heun.py code. Alternatively, one can use theosc_odespy.pycode by just providing the argumentodespy_methods= [odespy.RK4] to thecompare function. Derivation Thederivationof the4th-orderRunge-Kuttamethodcanbepresented in a pedagogical way that brings many fundamental elements of numerical dis- cretization techniques together and that illustrates many aspects of “numerical thinking”whenconstructingapproximatesolutionmethods.