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Calculate u1, u2, and u3.
1.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 the differential equation u = u with u(0) = 1. We use time steps Δt = 1. a) Start at t = 0 and draw a straight line with slope u…
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compare the three values u1, u2, and u3 with the values obtained in Exercise 8.2.
1.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 implementation of ode_FE. a) The solution computed by hand in Exercise 8.2 can be used as a reference solution. Make a function test_ode_FE_1()that calls ode_FE to compute three time…
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find through experimentation the largest value of Δt where the exact solution and the numerical solution by Heun’s method cannot be distinguished visually.
a) A second-order Runge-Kutta method, also known has Heun’s method, is derived in Sect. 8.4.5. Make a function ode_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 u = f (u, t), u(0) = U0, t ∈ (0, T ], with this method using…
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Calculate u1, u2, and u3. Check that the numbers are the same as obtained in a)-c).
1.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 the differential equation u = u with u(0) = 1. We use time steps Δt = 1. a) Start at t = 0 and draw a straight line with slope u…
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Make a function test_ode_FE_1()that calls ode_FE to compute three time steps in the problem u = u, u(0) = 1, and compare the three values u1, u2, and u3 with the values obtained in Exercise 8.2.
1.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 implementation of ode_FE. a) The solution computed by hand in Exercise 8.2 can be used as a reference solution. b) The test in a) can be made more general using…
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For the case in b), find through experimentation the largest value of Δt where the exact solution and the numerical solution by Heun’s method cannot be distinguished visually.
a) A second-order Runge-Kutta method, also known has Heun’s method, is derived in Sect. 8.4.5. Make a function ode_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 u = f (u, t), u(0) = U0, t ∈ (0, T ], with this method using…
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Write up the complete model, implement it, and rerun the case from Sect. 8.3.8 with various choices of parameters to illustrate various effects.
1.In the SIRV model with time-dependent vaccination from Sect. 8.3.9, we want to test the effect of an adaptive vaccination campaign where vaccination is offered 2.We consider the SIRV model from Sect. 8.3.8, but now the effect of vaccination is time-limited. After a characteristic period of time, π, the vaccination is no more effective and…
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Consider the file osc_FE.py implementing the Forward Euler method for the oscillating system model (8.43)–(8.44).
1.Consider the file osc_FE.py implementing the Forward Euler method for the oscillating system model (8.43)–(8.44). The osc_FE.py code is what we often refer to as a flat program, meaning that it is just one main program with no functions. Your task is to refactor the code in osc_FE.py according to the specifications below. Refactoring, means…
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Add a call to osc_energy in the programs osc_FE.py and osc_EC.py and plot the sum of the kinetic and potential energy.
a) Make a function osc_energy(u, v, omega) for returning the potential and kinetic energy of an oscillating system described by (8.43)–(8.44). The potential energy is taken as 1 2ω2u2 while the kinetic energy is 1 2 v2. (Note that these expressions are not exactly the physical potential and kinetic energy, since these would be 1…
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Find an expression for the Nn in terms of Nn−1 and formulate an algorithm for computing Nn, n = 1, 2,…,Nt .
1.We consider the ODE problem N (t) = rN(t), N(0) = N0. At some time, tn = nΔt, we can approximate the derivative N (tn) by a backward difference, see Fig. 8.22: N (tn) ≈ N(tn) − N(tn − Δt) Δt = Nn − Nn−1 Δt , which leads to Nn − Nn−1 Δt =…
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