$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\tp}{\thinspace .}
$$
Estimating the convergence rate \( r \)
Perform numerical experiments: \( (\Delta t_i, E_i) \), \( i=0,\ldots,m-1 \).
Two methods for finding \( r \) (and \( C \)):
- Take the logarithm of (1), \( \ln E = r\ln \Delta t + \ln C \),
and fit a straight line to the data points \( (\Delta t_i, E_i) \),
\( i=0,\ldots,m-1 \).
- Consider two consecutive experiments, \( (\Delta t_i, E_i) \) and
\( (\Delta t_{i-1}, E_{i-1}) \). Dividing the equation
\( E_{i-1}=C\Delta t_{i-1}^r \) by \( E_{i}=C\Delta t_{i}^r \) and solving
for \( r \) yields
$$
\begin{equation}
r_{i-1} = \frac{\ln (E_{i-1}/E_i)}{\ln (\Delta t_{i-1}/\Delta t_i)}
\tag{2}
\end{equation}
$$
for \( i=1,=\ldots,m-1 \).
Method 2 is best.
