Page - 155 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.4 Exercises 155 Write up the complete model, implement it, and rerun the case from Sect. 4.2.8 withvariouschoicesofparameters to illustratevariouseffects. Filename:SIRV1_V2S.py. Exercise4.8:Refactoraflatprogram Consider the fileosc_FE.py implementing the ForwardEulermethod for the os- cillating systemmodel (4.43)–(4.44). Theosc_FE.py iswhatweoften refer to as aflat program,meaning that it is just onemainprogramwithno functions. Toeas- ily reuse thenumericalcomputations inothercontexts,place thepart thatproduces thenumerical solution (allocationofarrays, initializing thearraysat timezero, and the time loop) in a functionosc_FE(X_0, omega, dt, T),which returnsu, v, t. Place theparticularcomputationalexample inosc_FE.py ina functiondemo(). Construct the file osc_FE_func.py such that the osc_FE function can easily be reused inotherprograms. InPython, thismeans thatosc_FE_func.pyis amodule that canbe imported in other programs. The requirementof amodule is that there shouldbenomainprogram,except in the test block. Youmust thereforecalldemo froma test block (i.e., theblockafterif __name__ == ’__main__’). Filename:osc_FE_func.py. Exercise4.9: SimulateoscillationsbyageneralODEsolver Solve the system (4.43)–(4.44) using the general solver ode_FE in the file ode_ system_FE.pydescribed in Sect. 4.2.6. Program theODEsystemand the call to theode_FE function inaseparatefileosc_ode_FE.py. Equip this file with a test function that reads a file with correctu values and compares these with those computed by the ode_FE function. To find correctu values,modifytheprogramosc_FE.py todumptheuarraytofile, runosc_FE.py, and let the test function read the referenceresults fromthatfile. Filename:osc_ode_FE.py. Exercise4.10:Compute theenergy inoscillations a) Make a function osc_energy(u, v, omega) for returning the potential and kineticenergyofanoscillatingsystemdescribedby(4.43)–(4.44).Thepotential energy is taken as 1 2 !2u2 while the kinetic energy is 1 2 v2. (Note that these expressionsarenotexactly thephysicalpotentialandkineticenergy,since these wouldbe 1 2 mv2 and 1 2 ku2 foramodelmx00CkxD0.) Place theosc_energy ina separatefileosc_energy.pysuch that the function canbecalled fromother functions. b) Addacall toosc_energyin theprogramsosc_FE.pyandosc_EC.pyandplot the sumof thekinetic andpotential energy. Howdoes the total energydevelop for theForwardEuler and theEuler-Cromerschemes? Filenames: osc_energy.py,osc_FE_energy.py,osc_EC_energy.py.