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Can someone help me verify if I'm differentiating this correctly?

$$\frac{\partial^2 u}{\partial x \, \partial y} = L_x L_y u$$ Choose to define $L_x$ as central difference $$L_x = \frac{u_{i+1,j}-u_{i-1,j}}{2 \Delta x}$$ $$L_y = \frac{u_{i,j+1}-u_{i,j-1}}{2 \Delta x}$$

$\dfrac{\partial^2 u}{\partial x \, \partial y}$ becomes: $$\frac{\partial^2 u}{\partial x \, \partial y} = (u_{i+1,j+1}-u_{i-1,j+1}-u_{i+1,j-1}+u_{i-1,j-1})\cdot\frac{1}{4 \, \Delta x \, \Delta y}$$

Is this correct?

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Yes, this is correct. The approximation is central step method. Reference : Click Here

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