$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Explanation of problems with Forward Euler
- \( a\Delta t= 2\cdot 1.25=2.5 \) and \( A=-1.5 \): oscillations and growth
- \( a\Delta t = 2\cdot 0.75=1.5 \) and \( A=-0.5 \): oscillations and decay
- \( \Delta t=0.5 \) and \( A=0 \): \( u^n=0 \) for \( n>0 \)
- Smaller \( Delta t \): qualitatively correct solution
