$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Discretization of derivatives at the boundary (1)
- How can we incorporate the condition \( u_x=0 \)
in the finite difference scheme?
- We used centeral differences for \( u_{tt} \) and \( u_{xx} \): \( \Oof{\Delta t^2, \Delta x^2} \) accuracy
- Also for \( u_t(x,0) \)
- Should use central difference for \( u_x \) to preserve second order accuracy
$$
\begin{equation}
\frac{u_{-1}^n - u_1^n}{2\Delta x} = 0
\tag{22}
\end{equation}
$$
