In numerical analysis, for the problems (mainly PDE) for which analytical solution is not available, the error computation can be done often with 'double mesh principle'.
For instant, Let $\mathcal U_{i,j}^{M,N}$ be the numerical approximation with $M$ and $N$ points in space and time directions respectively. Suppose we need error at the final time axis, Then the $L_{\infty}$ norm error (maximum absolute error, $e_{\max}^{M,N}$) is given by
$$e_{\max }^{M, N}=\max _{0 \leq m \leq M}\left|\mathcal U_{m,N}^{M,N}-\mathcal U_{2m,2N}^{2M,2N}\right|,$$
and the corresponding order of convergence is given by $$p_{\max}^{M,N}=\log_2\left(\frac{e_{\max}^{M,N}}{e_{\max}^{2M,2N}}\right),$$
On what logic the spatial order can be calculated by this formula using $\log$ function with base $2$? This formula has been used in several articles, but I am not sure about the justification behind it.
Best Answer
Suppose that the error satisfies the following asymptotic approximation:
$$\epsilon_M\sim\frac\alpha{M^\beta},\quad\alpha\in\mathbb R\land\beta\in\mathbb R_{>0}$$
This means we have:
$$\mathcal U_M-\mathcal U_\infty\sim\frac\alpha{M^\beta}$$
Plugging it into the first equation, we can then see that:
\begin{align}\mathcal U_M-\mathcal U_{2M}&=(\mathcal U_M-\mathcal U_\infty)-(\mathcal U_{2M}-\mathcal U_\infty)\\&\sim\frac\alpha{M^\beta}-\frac\alpha{(2M)^\beta}\\&=\left(1-\frac1{2^\beta}\right)\frac\alpha{M^\beta}\\&\sim\left(1-\frac1{2^\beta}\right)\epsilon_M\\&=\hat\epsilon_M\end{align}
is a rough approximation of $\epsilon_M$ (it's only off by a constant factor).
Plugging it into the second equation, we can see that:
\begin{align}\log_2\left(\frac{\hat\epsilon_M}{\hat\epsilon_{2M}}\right)&\approx\log_2\left(\frac{(1-2^{-\beta})\alpha M^{-\beta}}{(1-2^{-\beta})\alpha(2M)^{-\beta}}\right)\\&=\log_2\left(\frac1{2^{-\beta}}\right)\\&=\log_2\left(2^\beta\right)\\&=\beta\end{align}
which is in fact the order of convergence.