$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Detour: new notation
To make formulas less overloaded and the mathematics as close as
possible to computer code, a new notation is introduced:
- \( u^{(1)} \) means \( u^{n-1} \)
- In general: \( u^{(\ell)} \) means \( u^{n-\ell} \)
- \( u \) is the unknown (\( u^n \))
Nonlinear equation to solve in new notation:
$$
F(u) = \Delta t u^2 + (1-\Delta t)u - u^{(1)} = 0
$$
