Study guide: Intro to Computing with Finite Difference Methods

$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} $$

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