$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
The \( \theta \) rule
The \( \theta \) rule condenses a family of finite difference approximations
in time to one formula
- \( \theta=0 \) gives the Forward Euler scheme in time
- \( \theta=1 \) gives the Backward Euler scheme in time
- \( \theta=\half \) gives the Crank-Nicolson scheme in time
Applied to \( u_t=\dfc u_{xx} \):
$$ \frac{u^{n+1}_i-u^n_i}{\Delta t} =
\dfc\left( \theta \frac{u^{n+1}_{i+1} - 2u^{n+1}_i + u^{n+1}_{i-1}}{\Delta x^2}
+ (1-\theta) \frac{u^{n}_{i+1} - 2u^n_i + u^n_{i-1}}{\Delta x^2}\right)
$$
Matrix entries:
$$ A_{i,i-1} = -F_o\theta,\quad A_{i,i} = 1+2F_o\theta\quad,
A_{i,i+1} = -F_o\theta$$
Right-hand side:
$$ b_i = u^n_{i} + F_o(1-\theta)
\frac{u^{n}_{i+1} - 2u^n_i + u^n_{i-1}}{\Delta x^2}
$$
