$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Checking convergence rates
The next function estimates convergence rates, i.e., it
- performs \( m \) simulations with halved time steps: \( 2^{-k}\Delta t \), \( k=0,\ldots,m-1 \),
- computes the \( L_2 \) norm of the error,
\( E = \sqrt{\Delta t_i\sum_{n=0}^{N_t-1}(u^n-\uex(t_n))^2} \) in each case,
- estimates the rates \( r_i \) from two consecutive
experiments \( (\Delta t_{i-1}, E_{i-1}) \) and \( (\Delta t_{i}, E_{i}) \),
assuming \( E_i=C\Delta t_i^{r_i} \) and \( E_{i-1}=C\Delta t_{i-1}^{r_i} \):
