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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.

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up vote 1 down vote accepted

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.

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Thank you Federico. It is somewhat strange, but with the definition that you give, Equation 17 of the paper seems to be consistent. Another possibility is that since the paper was translated from Russian into English, there was an error made by the typesetter. I've tried to request the original version of the paper for comparison, since there is a possibility that something was left out. Moreover, I strongly suspect that the $u_{i,j + 2}^k$ term of Equation 17 should be $u_{i,j - 2}^k$, and that the $\{ C_2 , \ldots C_6 \}$ coefficients are of the wrong sign. –  Nicholas Kinar Aug 25 '10 at 2:40
    
It also appears that $C_1 = (1 + 7r)d$ and $d = (1 - 7r)^{ - 1}$ –  Nicholas Kinar Aug 25 '10 at 3:05
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