Page - 160 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

160 4 SolvingOrdinaryDifferentialEquations Filename:osc_BE.py. Exercise4.18: SetupaForwardEuler schemefornonlinearanddamped oscillations Derive a Forward Euler method for the ODE system (4.68)–(4.69). Compare themethod with the Euler-Cromer scheme for the sliding friction problem from Sect. 4.3.11: 1. Does theForwardEuler schemegivegrowingamplitudes? 2. Is theperiodofoscillationaccurate? 3. What is the required time step size for the twomethods to have visually coin- cidingcurves? Filename:osc_FE_general.py. Exercise4.19:Discretizean initial condition Assume that the initial condition onu0 is nonzero in the finite differencemethod fromSect. 4.3.12:u0.0/DV0. Derive the special formula foru1 in this case. Filename:ic_with_V_0.pdf. OpenAccess This chapter is distributedunder the termsof theCreativeCommonsAttribution- NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, duplication, adaptation, distribution and reproduction in anymediumor format, as long as you give appropriate credit to the original author(s) and the source, a link isprovided to theCreativeCommons licenseandanychangesmadeare indicated. The images or other third partymaterial in this chapter are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if suchmaterial is not included in thework’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce thematerial.