The central difference formulas are:
\begin{equation}
\frac{\partial ^2 u}{\partial x^2} \approx \frac{u_{i+1,j}-2u_{i,j} + u_{i-1,j}}{\Delta x^2}
\end{equation}
\begin{equation}
\frac{\partial ^2 u}{\partial x \partial y} \approx \frac{u_{i+1,j+1}-u_{i+1,j-1} - u_{i-1,j+1} +...