Page - 131 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.3 OscillatingOne-DimensionalSystems 131 u = zeros(N_t+1) v = zeros(N_t+1) # Initial condition u[0] = 2 v[0] = 0 # Step equations forward in time for n in range(N_t): v[n+1] = v[n] - dt*omega**2*u[n] u[n+1] = u[n] + dt*v[n+1] 4.3.5 The2nd-OrderRunge-KuttaMethod(orHeun’sMethod) A very popular method for solving scalar and vector ODEs of first order is the 2nd-orderRunge-Kuttamethod (RK2), also known asHeun’smethod. The idea, first thinkingofascalarODE, is to formacentereddifferenceapproximationto the derivativebetween two timepoints: u0.tnC 1 2 t/ u nC1 un t : Thecentereddifferenceformula isvisualized inFig.4.20. Theerror in thecentered difference is proportional to t2, oneorderhigher than the forwardandbackward differences,whichmeans that ifwehalve t, theerror ismoreeffectively reduced in thecentereddifferencesince it is reducedbya factorof four rather than two. Theproblemwith such a centered scheme for thegeneralODEu0 D f.u;t/ is thatweget unC1 un t Df.unC12 ;tnC12/; which leads to difficulties sincewe do not knowwhatunC 1 2 is. However,we can approximate the value off between two time levels by the arithmetic average of Fig.4.20 Illustrationof acentereddifferenceapproximation to thederivative