While reading some papers translated from the Russian literature, I've noticed that a delta symbol can be used to denote a FDTD stencil that discretizes a PDE. For example, in [1], a fourth order mixed partial derivative term is denoted by


$
2\frac{{\partial ^4 u}}{{\partial ^2 x\partial ^2 y}} = \Delta \_{xy}^4 u^{k + 1} \_{i + 1,j + 1} + \Delta \_{xy}^4 u^k \_{i - 1,j - 1} 
$


where an example is given of


$\Delta \_{xy}^4 u\_{i + 1,j + 1} = \Delta \_x^2 u\_{i + 1,j + 2} - 2\Delta \_x^2 u\_{i + 1,j + 1} + \Delta \_x^2 u\_{i + 1,j}$


Notice that this example given in the paper does not have the $\{ k,k + 1\} $ superscipts. 


Clearly ${i,j}$ are spatial indices and $k$ is the timestep. But what is being implied by the use of the delta symbol? I suspect that this is a differential, but I have never seen a differential with $u\_{i,j}$ and $i,j$ indices. The author does not define the symbol in his paper, so I think that it should be implicitly understood. I am also unsure as to whether such a notation has also been used by other authors. 


How would I write out $\Delta \_{xy}^4 u\_{i + 1,j + 1}$ and $\Delta \_{xy}^4 u\_{i - 1,j - 1}$ using a 5-point stencil or 7-point stencil? Are there any other papers which use similar notation?


[1] V. Saul'yev, “A difference method for solving parabolic equations of any order,” Computational Mathematics and Mathematical Physics, vol. 36(12), 1996, pp. 1697-1700.



Delta notation used for describing numerical stencil
If I saw that symbol alone, I would guess that $\Delta^2\_x u\_{i,j}:=u\_{i+1,j}-2u\_{i,j}+u\_{i-1,j}$. But with that definition the stencil $\Delta^4\_{xy}u\_{i+1,j+1}$ looks very odd.


