Page - 152 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

152 4 SolvingOrdinaryDifferentialEquations and inserting thefinitedifferenceapproximations tou00 andu0 results in m unC1 2unCun 1 t2 Cbu nC1 un 1 2 t Cs.un/DFn; (4.81) whereFn is a short notation forF.tn/. Equation (4.81) is linear in the unknown unC1, sowecaneasily solve for thisquantity: unC1 D 2munC b 2 t m un 1C t2.Fn s.un// mC b 2 t 1 : (4.82) As in thecasewithoutdamping,weneed toderiveaspecial formula foru1. The initial conditionu0.0/ D 0 implies also now thatu 1 D u1, andwith (4.82) for nD0,weget u1 Du0C t 2 2m .F0 s.u0//: (4.83) In themoregeneral casewithanonlineardamping termf.u0/, mu00Cf.u0/Cs.u/DF.t/; weget m unC1 2unCun 1 t2 Cf unC1 un 1 2 t Cs.un/DFn; which isanonlinearalgebraicequation forunC1 thatmustbe solvedbynumerical methods.Amuchmoreconvenientschemearises fromusingabackwarddifference foru0, u0.tn/ u n un 1 t ; because thedamping termwill thenbeknown, involvingonlyun andun 1, andwe caneasily solve forunC1. The downside of the backward difference compared to the centered difference (4.80) is that it reduces the order of the accuracy in the overall scheme from t2 to t. In fact, the Euler-Cromer scheme evaluates a nonlinear damping term as f.vn/whencomputingvnC1,andthisisequivalenttousingthebackwarddifference above. Consequently, the convenience of the Euler-Cromer scheme for nonlinear dampingcomesat a cost of lowering theoverall accuracyof the scheme fromsec- ond to first order in t. Using the same trick in the finite difference scheme for the second-orderdifferential equation, i.e., using thebackwarddifference inf.u0/, makes this schemeequallyconvenientandaccurateas theEuler-Cromerscheme in thegeneralnonlinearcasemu00Cf.u0/Cs.u/DF .