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…

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…

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…

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…

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 =…

1.It is recommended to do Exercise 8.12 prior to the present one. Here we look at the same population growth model N (t) = rN(t), N(0) = N0. The time derivative N (t) can be approximated by various types of finite differences. Exercise 8.12 considers a backward difference (Fig. 8.22), while Sect. 8.2.2 explained the…

1.A general ODE problem u (t) = f (u, t), u(0) = U0, may be solved numerically by the third order Runge-Kutta method. The computational scheme reads un+1 = un + Δt 6 (k1 + 4k2 + k3) , n = 0, 1,…,Nt − 1, k1 = f (un, tn), k2 = f (un +…

1.Differing from the single-step methods presented in this chapter, we have the multistep methods, for example the Adams-Bashforth methods. With the single-step methods, un+1 is computed by use of the solution from the previous time step, i.e. un. In multi-step methods, the computed solutions from several previous time steps, e.g., un, un−1 and un−2 are…

1.Consider (8.43)–(8.44) modeling an oscillating engineering system. This 2×2 ODE system can be solved by the Backward Euler scheme, which is based on discretizing derivatives by collecting information backward in time. More specifically, u (t) is approximated as u (t) ≈ u(t) − u(t − Δt) Δt . A general vector ODE u = f…

1.Let y be a scalar function of time t and consider the nonlinear ODE y + y = ty3, t ∈ (0, 4), y(0) = 1 2 . a) Assume you want to solve this ODE numerically by the Backward Euler method. Derive the computational scheme and show that (contrary to the Forward Euler scheme)…