$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Computing the coefficient between mesh points
- Given \( q(x) \): compute \( q_{i+\half} \) as \( q(x_{i+\half}) \)
- Given \( q \) at the mesh points: \( q_i \), use an average
$$
\begin{align}
q_{i+\half} &\approx
\half\left( q_{i} + q_{i+1}\right) =
[\overline{q}^{x}]_i
\quad &\hbox{(arithmetic mean)}
\tag{26}\\
q_{i+\half} &\approx
2\left( \frac{1}{q_{i}} + \frac{1}{q_{i+1}}\right)^{-1}
\quad &\hbox{(harmonic mean)}
\tag{27}\\
q_{i+\half} &\approx
\left(q_{i}q_{i+1}\right)^{1/2}
\quad &\hbox{(geometric mean)}
\tag{28}
\end{align}
$$
The arithmetic mean in (26) is by
far the most used averaging technique.
