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 forward difference (Fig. 8.4). A centered difference is more accurate than a backward or forward difference: N
(tn + 1 2 Δt) ≈ N(tn + Δt) − N(tn) Δt = Nn+1 − Nn Δt . This type of difference, applied at the point t n+1 2 = tn + 1 2Δt, is illustrated geometrically in Fig. 8.23.

a) Insert the finite difference approximation in the ODE N
= rN and solve for the unknown Nn+1, assuming Nn is already computed and hence known. The resulting computational scheme is often referred to as a Crank-Nicolson scheme.

b) Implement the algorithm in a) in a function growth_CN(N_0, dt, T) for solving N
= rN, N(0) = N0, t ∈ (0, T ], with time step Δt (dt).

c) Make plots for comparing the Crank-Nicolson scheme with the Forward and Backward Euler schemes in the same test problem as in Exercise 8.12.