$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Verification via trivial solutions
- Start debugging of a new code with trying a problem
where \( u=\hbox{const} \neq 0 \).
- Choose \( u=C \) (a constant). Choose any \( a(t) \) and set
\( b=a(t)C \) and
\( I=C \).
- "All" numerical methods will reproduce \( u=_{\hbox{const}} \)
exactly (machine precision).
- Often \( u=C \) eases debugging.
- In this example: any error in the formula for \( u^{n+1} \)
make \( u\neq C \)!
