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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):

enter image description here

Inserting lines 2 and 3 into 1 gives (according to the pdf) the following:

enter image description here

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.

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Section 2.5 of this PDF document goes through some additional details and the error does in fact work out to be second order. The key is that the term multiplying the $\Delta y^2/\Delta x$ term is equal to

$$\Delta x\left( \dfrac{\partial^4u}{\partial x\partial y^3} +\mathcal{O}(\Delta x^2) \right)$$

The $1/\Delta x$ cancels the $\Delta x$ in this term and the error is formally second order in $\Delta x$ and $\Delta y$ independently.

You can find other resources by searching for "cross derivatives" and "finite difference"

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The discretization presented is of second order accuracy. To look at it intuitively, both the first derivatives in x and y are taken using central difference, which puts their order of accuracy at 2. So, a combination of these two 2nd order central difference terms should atleast be 2nd order accurate.

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  • $\begingroup$ Could you elaborate on why a combination of two 2nd-order centred difference approximations should be intuitively 2nd-order? This seems non-obvious to me. $\endgroup$ – hertzsprung May 17 '16 at 10:32
  • $\begingroup$ When you have a combination of terms in an expression, the order of accuracy of the whole expression is given by the order of individual term with lowest order. Here, we have two terms with 2nd order accuracy, hence the order of accuracy of the whole expression should be of second order. Mathematically, the order of accuracy of a term is the trailing $ O(x^n) $. When you add these up, the one with the lowest order remains and becomes the order of whole expression. Hope this helps. $\endgroup$ – Ravi May 19 '16 at 15:48

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