Page - 254 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

254 8 SolvingOrdinaryDifferentialEquations Fig. 8.26 Comparison of the Runge-Kutta-Fehlberg adaptive method against the Euler-Cromer scheme for a long timesimulation (200 periods) 8.4.7 TheFourth-OrderRunge-KuttaMethod The fourth-order Runge-Kutta method (RK4) is clearly the most widely used methodtosolveODEs. Itspowercomesfromhighaccuracyevenwithnot sosmall timesteps. The Algorithm We first just state the four-stagealgorithm: un+1 =un+Δt 6 ( fn+2fˆn+12 +2f˜n+12 + f¯n+1 ) , (8.59) where fˆn+ 1 2 =f(un+ 1 2 Δtfn,tn+12), (8.60) f˜n+ 1 2 =f(un+ 1 2 Δtfˆn+ 1 2,t n+12), (8.61) f¯n+1 =f(un+Δtf˜n+12,tn+1). (8.62) Application Wecanrunthesamesimulationas inFigs.8.19,8.21,and8.24,for40 periods. The 10 last periods are shown in Fig. 8.27. The results look as impressive as thoseof the Euler-Cromermethod.