1

I am used to deriving consistency for approximations of derivatives such as

$\frac{\partial u(x,y)}{\partial x} = \frac{u(x+h,y) - u(x,y)}{h}$

and then going on with taylor series:

$u(x+h,y) = u(x,y) + hu'(x,y) + \frac{h^2}{2}u''(x,y) ... $

but how does this work for this for example:

$\frac{\partial^2 u(x,y)}{\partial x \partial y} = \frac{u(x+h,y+h) - u(x-h,y+h) - u(x+h, y-h) + u(x-h,y-h)}{4h^2}$

where I have $u(x+h,y+h)$? What would be the taylor series of this?

binaryBigInt
  • 404

1 Answers1

0

$u(x+h, y+h) = u(x,y) + h\frac{\partial u}{\partial x}(x,y) + h\frac{\partial u}{\partial y} (x,y) + \frac{h^2}{2}\frac{\partial^2}{\partial x^2}u(x,y) + \frac{h^2}{2}\frac{\partial^2 }{\partial y^2}u(x,y) + \frac{h^2}{2}\frac{\partial^2}{\partial x \partial y} u(x,y) + \frac{h^2}{2}\frac{\partial^2}{\partial y \partial x} u(x,y) $

binaryBigInt
  • 404