$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Arithmetic means: which variant is best?
Is there any differences in accuracy between
- two factors of arithmetic means
- the arithmetic mean of a product
More precisely,
$$
\begin{align*}
[PQ]^{n+\frac{1}{2}} = P^{n+\frac{1}{2}}Q^{n+\frac{1}{2}}
&\approx
\frac{1}{2}(P^n + P^{n+1})\frac{1}{2}(Q^n + Q^{n+1})\\
[PQ]^{n+\frac{1}{2}} & \approx \frac{1}{2}(P^nQ^n + P^{n+1}Q^{n+1})
\end{align*}
$$
It can be shown (by Taylor series around \( t_{n+\frac{1}{2}} \)) that
both approximations are \( \Oof{\Delta t^2} \)
