$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Using an analytical solution of physical significance
- Standing waves occur in real life on a string
- Can be analyzed mathematically (known exact solution)
$$
\begin{equation}
\uex(x,y,t)) = A\sin\left(\frac{\pi}{L}x\right)
\cos\left(\frac{\pi}{L}ct\right)
\tag{19}
\end{equation}
$$
- PDE data: \( f=0 \), boundary conditions
\( \uex(0,t)=\uex(L,0)=0 \), initial
conditions \( I(x)=A\sin\left(\frac{\pi}{L}x\right) \) and \( V=0 \)
- Note: \( u_i^{n+1}\neq\uex(x_i,t_{n+1} \), and we do not know
the error, so testing must aim at reproducing the expected
convergence rates
