Given is the following difference scheme:
$L_h u = \frac{u_{i-1,j} - 2 u_{i,j} + u_{i+1,j}}{h^2} + \frac{u_{i,j-1} - 2 u_{i,j} + u_{i,j+1}}{h^2}$.
This can be written in the form
$L_h u = \sum_{k,l=1}^{3} \sigma_{l,k} u_{i+2-k,j+2-l}$
with
$\sigma = \pmatrix{0 & 1 & 0\\1 & -4 & 1\\0 & 1 & 0}$.
Let be $a^{+} = \text{max}(\{a,0\})$ and $a^{-} = \text{min}(\{a,0\})$ as well as
$\tau = \pmatrix{-a_{12}^{-} & a_{22} - |a_{12}| & a_{12}^{+}\\a_{11}-|a_{12}| & 2 (|a_{12}|-a_{11}-a_{22}) & a_{11}-|a_{12}|\\a_{12}^{+} & a_{22}-|a_{12}| & -a_{12}^{-}}$.
My question:
Which difference operator is approximated by $L_h = \sum_{k,l=1}^{3} \tau_{l,k} u_{i+2-k,j+2-l}$ and with wich order?
My idea:
Using the difference schemes of the first and second derivations of $u$. Unfortunately, I didn't got far with this idea.
