Computing Mixed Derivatives by Finite Differences

The post is aimed to summarize various finite difference schemes for partial derivatives estimation dispersed in comments on the Central Differences page. To gather them all in one place as a reference.

Listed formulas are selected as being advantageous among others of similar class – highest order of approximation, low rounding errors, etc. Please use comments to add other schemes.

Second order


(1)   \begin{equation*} \displaystyle{{\frac{\partial^2{f}}{\partial{x}\partial{y}}}\approx \frac{1}{4\,h^2}\left[f_{-1,-1}+f_{1,1}-f_{1,-1}-f_{-1,1}\right]} \end{equation*}


(2)   \begin{equation*} \frac{\partial^2 f}{\partial x \partial y} \approx \frac{1}{144 h^2}\left[   \begin{array}{l}     8(f_{1,-2}+f_{2,-1}+f_{-2,1}+f_{-1,2})-8(f_{-1,-2}+f_{-2,-1}+f_{1,2}+f_{2,1})\\    -(f_{2,-2}+f_{-2,2}-f_{-2,-2}-f_{2,2})+64(f_{-1,-1}+f_{1,1}-f_{1,-1}-f_{-1,1})   \end{array}\right] \end{equation*}

The last formula is provided by “Felix”.

There are other formulas (e.g. provided by Edward F. Valeev) which can be beneficial for some applications (e.g. to use minimum number of function values in trade-off with decreased accuracy).

Third & Fourth order

To be continued…

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