Chapter D1.2, Problem 50E

Stability of Euler’s method Consider the initial value problem y'(t) = –ay, y(0) = 1, where a > 0; it has the exact solution y(t) = e^–at, which is a decreasing function.

a.    Show that Euler’s method applied to this problem with time step h can be written u₀ = 1, u_k+l = (1 – ah)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.    Explain why as k increases the Euler approximations u_k = (1 – ah)^k decrease in magnitude only if |1 – ah| < 1.

d.    Show that the inequality in part (c) implies that the time step must satisfy the condition $0<h<\frac{2}{a}$. If the time step does not satisfy this condition, then Euler’s method is unstable and produces approximations that actually increase in time.