Page - 309 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

9.3 Exercises 309 Remarks If the interest is in the stationary limit of a diffusion equation, one can either solve the associated Laplace orPoisson equationdirectly,or use a Backward Euler schemefor the time-dependentdiffusionequationwith a very long time step. Using a Forward Euler scheme with small time steps is typically inappropriate in such situations because the solution changes more and more slowly, but the time step must still be kept small, and it takes“forever” to approach the stationarystate. This is yet another example why one needs implicit methods like the Backward Euler scheme. Exercise9.10:Solvea Two-PointBoundaryValue Problem Solve the following two-pointboundary-valueproblem u′′(x)=2, x∈ (0,1), u(0)=0, u(1)=1 . Hint DoExercise9.9.Modifytheboundaryconditioninthecodesoit incorporates aknownvalueforu(1). Filename:2ptBVP.py. Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation,distributionandreproduction inanymediumor format, as longasyougiveappropriate credit to the original author(s) and thesource, provide a link to theCreative Commons licenceand indicate ifchanges were made. The images or other third party material in this chapter are included in the chapter’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons licence and your intended use is not permitted by statutory regulation orexceeds thepermitted use, you willneed toobtain permission directly from thecopyright holder.