Mixed derivatives

A finite difference approximations for the mixed partial derivatives one get in the same way. For example, let us find the central approximation for the derivative

$$\frac{\partial^2u}{\partial x\partial y}=\frac{\partial}{\partial... ...riangle y)-u(x,y-\triangle y)}{2\triangle y}+\mathcal{O}(\triangle y^2)\biggr)=$$

$$=\frac{u(x+\triangle x,y+\triangle y)-u(x-\triangle x,y+\triangle... ...-\triangle y)}{4\triangle x\triangle y}+\mathcal{O}(\triangle x^2\triangle y^2)$$