Study guide: Intro to Computing with Finite Difference Methods

Loading [MathJax]/extensions/TeX/boldsymbol.js

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

Computation of the truncation error

The residual $R^n$ is the truncation error.

How does $R^n$ vary with $\Delta t$ ?

Tool: Taylor expand

$\uex$ around the point where the ODE is sampled (here

$t_n$ )

$\uex(t_{n+1}) = \uex(t_n) + \uex'(t_n)\Delta t + \half\uex''(t_n) \Delta t^2 + \cdots$ Inserting this Taylor series in (31) gives

$R^n = \uex'(t_n) + \half\uex''(t_n)\Delta t + \ldots + a\uex(t_n)$ Now,

$\uex$ solves the ODE

$\uex'=-a\uex$ , and then

$R^n \approx \half\uex''(t_n)\Delta t$ This is a mathematical expression for the truncation error.