Page - 179 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

5.2 Exercises 179 TheOdespypackageoffers thismethodasodespy.Backward2Step.Thepurpose of this exercise is tocompare threemethodsandanimate the threesolutions: 1. TheBackwardEulermethodwith t D0:001 2. Thebackward2-stepmethodwith t D0:001 3. Thebackward2-stepmethodwith t D0:01 Choose themodelproblemfromSect. 5.1.4. Filename:rod_BE_vs_B2Step.py. Exercise5.4:Exploreadaptiveand implicitmethods Weconsider the sameproblemas inExercise5.2. Nowwewant to explore theuse ofadaptiveand implicitmethods fromOdespy to see if theyaremoreefficient than the Forward Euler method. Assume that youwant the accuracy provided by the ForwardEulermethodwith itsmaximum t value. Since thereexists ananalytical solution,youcancomputeanerrormeasure that summarizes theerror in spaceand timeover thewhole simulation: E D s x t X i X n .Uni uni/2 : Here,Uni is theexactsolution.UsetheOdespypackagetorunthefollowingimplicit andadaptivesolvers: 1. BackwardEuler 2. Backward2Step 3. RKFehlberg Experiment to see if you can use larger time steps than what is required by the ForwardEulermethodandget solutionswith the sameorderofaccuracy. Hint Toavoidoscillations in thesolutionswhenusingtheRKFehlbergmethod, the rtol andatolparameters toRKFFehlbergmust be set no larger than 0.001 and 0.0001,respectively.Youcanprintoutsolver_RKF.t_alltoseeall thetimesteps usedbytheRKFehlbergsolver(ifsolver is theRKFehlbergobject).Youcanthen compare thenumberof timestepswithwhat is requiredby theothermethods. Filename:ground_temp_adaptive.py. Exercise5.5: Investigate the rule a) The Crank-Nicolsonmethod for ODEs is very popular when combined with diffusionequations. For a linearODEu0 Dau it reads unC1 un t D 1 2 .aunCaunC1/: Apply the Crank-Nicolson method in time to the ODE system for a one- dimensionaldiffusionequation. Identify the linear system tobesolved.