Step 1: Discretizing the domain

The time domain $[0,T]$ is represented by a mesh: a finite number of $N_t+1$ points $$0 = t_0 < t_1 < t_2 < \cdots < t_{N_t-1} < t_{N_t} = T$$

• We seek the solution $u$ at the mesh points: $u(t_n)$, $n=1,2,\ldots,N_t$.
• Note: $u^0$ is known as $I$.
• Notational short-form for the numerical approximation to $u(t_n)$: $u^n$
• In the differential equation: $u$ is the exact solution
• In the numerical method and implementation: $u^n$ is the numerical approximation, $\uex(t)$ is the exact solution