Background

The dG0 timestep method (backward Euler) for the initial value problem

$$\begin{array}{rcl} \dot u (t) &=& f(u(t)) \quad 0<t<T \\ u(0) &=& u^0 \end{array}$$ (1)

is, Find $U(t_n)$ successively for $n=0,\dots, N$ according to

$$U(t_n) = U(t_{n-1}) + k_n f(U(t_n))$$ (2)

This is a nonlinear equation if $f$ is nonlinear in $u$.

Before today's computer session, make sure that you understand and can answer the following question.

Question 1 Derive the equation for the backward Euler method (2) from the IVP (1).