Page - 255 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.4 Oscillating1DSystems:ASecondOrderODE 255 Fig. 8.27 The last 10 of 40 periods of oscillationsby the fourth-order Runge-Kutta method Implementation The stages in the fourth-order Runge-Kutta method can easily be implemented as a modification of the osc_Heun.py code. Alternatively, one can use the osc_odespy.py code by just providing the argument odespy_methods=[odespy.RK4] to thecompare function. Derivation The derivation of the fourth-order Runge-Kutta method can be pre- sented in a pedagogical way that brings many fundamental elements of nu- merical discretization techniques together. It also illustrates many aspects of the “numerical thinking” required for constructing approximate solution meth- ods. We start with integrating thegeneralODEu′ =f(u,t)overa timestep, from tn to tn+1, u(tn+1)−u(tn)= tn+1∫ tn f(u(t),t)dt . Thegoalofthecomputationisu(tn+1) (writtenun+1),whileu(tn) (writtenun) is the mostrecentlyknownvalueofu.Thechallengewith theintegral is that the integrand involves the unknownubetween tn and tn+1.