$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
Analysis of the finite difference schemes
Stability:
- \( |A| < 1 \): decaying numerical solutions (as we want)
- \( A < 0 \): oscillating numerical solutions (as we do not want)
Accuracy:
- Compare numerical and exact amplification factor: \( A \) vs \( \Aex = \exp{(-\dfc k^2 \Delta t)} \)
