The problem is to find a 2nd order finite difference approximation of the partial derivative uxy, where u is a function of x and y.
Page 5 of this pdf I found does a centered difference approximation it in two steps. It first does the 2nd order centered finite-difference approximation of one of the partials, and then inserts the approximation of the second partial into it (using the same formula):
Inserting lines 2 and 3 into 1 gives (according to the pdf) the following:
The last O[(Δ x)2,(Δ y)2] is what I have a problem with. Notice that when the O(Δ y)2 terms of lines 2 and 3 go into the numerator of 1, they are being divided by the Δ x in the denominator. So how come the residual terms in line 3 are of O(Δ y)2 instead of O(Δ y2/Δ x)? Would this be a '2nd order' approximation any more? (If, say, grid-spacing along both axes are the same (Δ x = Δ y = h), the term is of order h2/h = h, not h2.)
My suggestion would be to use a higher order approximation (3rd or more) in lines 2 and 3 in order to survive the division by Δ x and still have the final expression in 2nd order. But I may be missing something here.