How to find the numerical error when we don't know the exact solution? When some quantity $x$ (e.g., the values of a solution of a PDE, using a finite difference method) is calculated numerically, we get its approximate value $x^*$. The error is $|x-x^*|$. But since we don't know $x$ itself, how is it possible to find the rrror?  
 A: Typically, the accuracy of the numerical approximation hinges on a parameter $h>0$ such as the time step when integration ordinary differential equations or the spacing between grid points when solving the Laplace equation on a square using the standard five point finite difference stencil with a fixed step size. You will frequently have an asymptotic error expansion of the form
\begin{equation}
T - A_h = \alpha h^p + \beta h^q + O(h^r), \quad p < q < r.
\end{equation}
Here $T$ is your target, i.e. the number which you which to compute, $A_h$ is the approximation obtained used the parameter $h$, $\alpha$ and $\beta$ are constants which depend on the target, but are independent of $h$ and the numbers $p < q < r$ reflect the properties of you method. You can estimate the principal error term, i.e. $\alpha h^p$ as follows
\begin{equation}
\alpha h^p \approx \frac{A_h - A_{2h}}{2^p - 1}
\end{equation}
provided that $h$ is sufficiently small. This can be judged by evaluating the fraction
\begin{equation}
F_h = \frac{A_{2h}- A_{4h}}{A_h - A_{2h}}
\end{equation}
which tends to $2^p$ and is close to $2^p$ precisely when the above approximation is good.
