Page - 305 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

9.3 Exercises 305 3. Thebackward2-stepmethodwithΔt=0.01 Choose the modelproblemfromSect.9.2.4. Filename:rod_BE_vs_B2Step.py. Exercise9.4:ExploreAdaptiveand Implicit Methods WeconsiderthesameproblemasinExercise9.2.Nowwewant toexploretheuseof adaptiveandimplicitmethodsfromOdespytosee if theyaremoreefficient than the ForwardEulermethod.AssumethatyouwanttheaccuracyprovidedbytheForward Eulermethodwith its maximumΔt value.Since thereexistsananalytical solution, youcancomputeanerrormeasure that summarizes theerror inspaceand timeover thewholesimulation: E= √ ΔxΔt ∑ i ∑ 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 thesameorderofaccuracy. Hint Toavoidoscillationsin thesolutionswhenusingtheRKFehlbergmethod, the rtol andatolparameters toRKFFehlbergmust be set no larger than 0.001 and 0.0001,respectively.Youcanprintoutsolver_RKF.t_alltoseeall the timesteps usedbytheRKFehlbergsolver(ifsolver is theRKFehlbergobject).Youcanthen compare thenumberof timestepswithwhat is requiredby theothermethods. Filename:ground_temp_adaptive.py. Exercise9.5: Investigate theθ Rule a) The Crank-Nicolson method for ODEs is very popular when combined with diffusionequations.Fora linearODEu′ =au it reads un+1 −un Δt = 1 2 (aun+aun+1). Apply the Crank-Nicolson method in time to the ODE system for a one- dimensionaldiffusionequation. Identify the linear system tobesolved.