Seite - 153 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.4 Exercises 153 4.4 Exercises Exercise4.1:Geometricconstructionof theForwardEulermethod Section 4.1.4 describes a geometric interpretation of the Forward Euler method. This exercisewill demonstrate thegeometric constructionof the solution indetail. Consider thedifferential equationu0 Duwithu.0/D 1. Weuse time steps t D 1. a) Start at t D 0 and drawa straight linewith slopeu0.0/ D u.0/ D 1. Go one timestep forward to t D t andmark the solutionpointon the line. b) Drawastraight line throughthesolutionpoint. t;u1/withslopeu0. t/Du1. Goone timestep forward to t D2 t andmark the solutionpointon the line. c) Drawastraight line through the solutionpoint .2 t;u2/with slopeu0.2 t/D u2. Go one time step forward to t D 3 t andmark the solution point on the line. d) Setup theForwardEuler schemefor theproblemu0 Du. Calculateu1,u2, and u3. Check that thenumbersare the sameasobtained ina)-c). Filename:ForwardEuler_geometric_solution.py. Exercise4.2:Make test functions for theForwardEulermethod Thepurposeof this exercise is tomake afiletest_ode_FE.py thatmakes use of the ode_FE function in the file ode_FE.py and automatically verifies the imple- mentationofode_FE. a) The solution computed by hand inExercise 4.1 can be used as a reference so- lution.Makea functiontest_ode_FE_1() that callsode_FE to compute three time steps in the problemu0 D u,u.0/ D 1, and compare the threevaluesu1, u2, andu3with thevaluesobtained inExercise4.1. b) Thetest ina)canbemademoregeneralusingthefact that iff is linear inuand does not dependon t, i.e.,wehaveu0 D ru, for someconstant r, theForward EulermethodhasaclosedformsolutionasoutlinedinSect.4.1.1:un DU0.1C r t/n. Use this result to construct a test functiontest_ode_FE_2() that runs anumberofsteps inode_FEandcomparesthecomputedsolutionwiththelisted formula forun. Filename:test_ode_FE.py. Exercise4.3: ImplementandevaluateHeun’smethod a) A2nd-orderRunge-Kuttamethod,alsoknownhasHeun’smethod, isderivedin Sect. 4.3.5.Makea functionode_Heun(f, U_0, dt, T) (as a counterpart to ode_FE(f, U_0, dt, T) in ode_FE.py) for solving a scalar ODE problem u0 D f.u;t/,u.0/ D U0, t 2 .0;T , with thismethod using a time step size t. b) Solve the simpleODEproblemu0 D u,u.0/ D 1, by theode_Heun and the ode_FE function. Make a plot that comparesHeun’smethod and the Forward Eulermethodwith the exact solutionu.t/ D et for t 2 Œ0;6 . Use a time step t D0:5.