Chapter D1.2, Problem 49E

Convergence of Euler’s method Suppose Euler’s method is applied to the initial value problem $y^{\prime }\left(t\right)=ay,y\left(0\right)=1$, which has the exact solution y(t) = e^at. For this exercise, let h denote the time step (rather than Δt). The grid points are then given by t_k = kh. We let u_k be the Euler approximation to the exact solution $y\left(t_k\right)$, for k = 0, 1, 2,….

a.    Show that Euler’s method applied to this problem can be written $u_0=1,u_{k+1}=\left(1+ah\right)u_k$, for k = 0, 1, 2, ….

b.    Show by substitution that u_k = (1 + ah)^k is a solution of the equations in part (a), for k = 0, 1, 2, ….

c.    Recall that $\underset{h\to 0}{\lim }{\left(1+ah\right)}^{a/h}=e^a$. Use this fact to show that as the time step goes to zero (h → 0. with t_k = kh fixed), the approximations given by Euler's method approach the exact solution of the initial value problem; that is, $\underset{h\to 0}{\lim }{u}_k=\underset{h\to 0}{\lim }{\left(1+ah\right)}^k=y\left(t_k\right)=e^{a{t}_k}.$