Page - 273 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.6 Exercises 273 corresponding lists, so that convergence rates (r) can be computed after the loop. Observe, in the loop, how ode_FE returns (the solution u and) the time mesh t, which then is used as input to the u_exact function, causing the exact function values to alsobecomputedat the verysamemeshpointsasu. The Forward Euler method is a first order method, so we should get r near 1 as thetimestepbecomessmallenough.Acall to this test functiondoesindeedconfirm (remove# in frontofprint(r)) thatrcomesveryclose to 1asdtgets smaller. 8.6 Exercises Exercise8.1:RestructureaGiven Code Section 8.1.1 gives a code for computing the development of water volume V in a tank. Restructure the code by introducing an appropriate function compute_V that computes and returns the volumes, and a functionapplication that calls the formerfunctionandplots the result. Note that your restructuring should not cause any change in program behavior, asexperiencedbyauser of the program. Filename:restruct_tank_case1.py. Exercise8.2:GeometricConstructionof the ForwardEuler Method Section 8.2.4 describes a geometric interpretation of the Forward Euler method. This exercise will demonstrate the geometric construction of the solution in detail. Consider thedifferentialequationu′ =uwithu(0)=1.Weuse timestepsΔt=1. a) Start at t=0anddrawa straight linewith slopeu′(0)=u(0)=1.Goone time step forward to t=Δt andmark thesolutionpointon the line. b) Drawastraight line throughthesolutionpoint (Δt,u1)withslopeu′(Δt)=u1. Goone timestep forward to t=2Δt andmark thesolutionpointon the line. c) Draw a straight line through the solution point (2Δt,u2)with slopeu′(2Δt)= u2.Goonetimestepforwardto t=3Δt andmarkthesolutionpointonthe line. d) Set up the Forward Euler scheme for the problemu′ =u. Calculateu1,u2, and u3.Check that the numbersare the sameasobtained ina)-c). Filename:ForwardEuler_geometric_solution.py. Exercise8.3:Make Test Functions for theForwardEuler Method The purpose of this exercise is to make a file test_ode_FE.py that makes use of the ode_FE function in the file ode_FE.py and automatically verifies the implementationofode_FE. a) The solution computed by hand in Exercise 8.2 can be used as a reference solution.Makeafunctiontest_ode_FE_1()thatcallsode_FE tocomputethree timesteps in theproblemu′ =u,u(0)=1,andcomparethe threevaluesu1,u2, andu3 with the valuesobtained in Exercise8.2. b) The test in a) can be made more general using the fact that if f is linear in u and does not depend on t, i.e., we have u′ = ru, for some constant r, the ForwardEulermethodhasaclosedformsolutionasoutlinedinSect.8.2.1:un= U0(1+rΔt)n.Use this result toconstructa test functiontest_ode_FE_2()that