$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Summary of observations
The characteristics of the displayed curves can be summarized as follows:
- The Backward Euler scheme always gives a monotone solution, lying above
the exact curve.
- The Crank-Nicolson scheme gives the most accurate results, but for
\( \Delta t=1.25 \) the solution oscillates.
- The Forward Euler scheme gives a growing, oscillating solution for
\( \Delta t=1.25 \); a decaying, oscillating solution for \( \Delta t=0.75 \);
a strange solution \( u^n=0 \) for \( n\geq 1 \) when \( \Delta t=0.5 \); and
a solution seemingly as accurate as the one by the Backward Euler
scheme for \( \Delta t = 0.1 \), but the curve lies below the exact
solution.
