Page - 143 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.3 OscillatingOne-DimensionalSystems 143 Anymethod for a systemoffirst-orderODEscanbeused to solve foru.t/ and v.t/. The Euler-Cromer scheme An attractive choice from an implementational, ac- curacy, and efficiency point of view is the Euler-Cromer schemewhere we take a forwarddifference in (4.68)andabackwarddifference in (4.69): vnC1 vn t D 1 m .F.tn/ s.un/ f.vn//; (4.70) unC1 un t DvnC1; (4.71) Wecaneasily solve for thenewunknownsvnC1 andunC1: vnC1 DvnC t m .F.tn/ s.un/ f.vn//; (4.72) unC1 DunC tvnC1 : (4.73) RemarkontheorderingoftheODEs The ordering of the ODEs in the ODE system is important for the extended model (4.68)–(4.69). Imagine thatwewrite theequationforu0first and then the one forv0. TheEuler-Cromermethodwould thenfirst use a forwarddifference for unC1 and then a backward difference for vnC1. The latter would lead to anonlinearalgebraicequation forvnC1, vnC1C t m f.vnC1/DvnC t m F.tnC1/ s.unC1/ ; iff.v/ is anonlinear functionofv. Thiswould requireanumericalmethod for nonlinear algebraic equations to find vnC1, while updatingvnC1 through a for- ward difference gives an equation forvnC1 that is linear and trivial to solve by hand. Thefileosc_EC_general.pyhasafunctionEulerCromerthat implementsthis method: def EulerCromer(f, s, F, m, T, U_0, V_0, dt): from numpy import zeros, linspace N_t = int(round(T/dt)) print ’N_t:’, N_t t = linspace(0, N_t*dt, N_t+1) u = zeros(N_t+1) v = zeros(N_t+1) # Initial condition u[0] = U_0 v[0] = V_0