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Is there any difference in superscript and subscript notation in the finite difference method? I see the same paper use (superscript for $x$ and superscript for $y$ notation) and (subscript for x and y notation)

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  • $\begingroup$ Please do not post images of math. Use MathJax. $\endgroup$ – G. Smith May 24 at 17:54
  • $\begingroup$ I see the same paper use (super subscript for $x$ and superscript for $y$ notation) and (subscript for $x$ and $y$ notation). That’s not true. Those equations consistently use subscripts for spatial indices and superscripts for temporal indices. And did you mean to write “super subscript”? $\endgroup$ – G. Smith May 24 at 19:32
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There's no way to know for sure with just these images posted, but most likely this is just a difference in convention between two sources or a difference in convention from a single source for how indices for time are printed vs. indices for spatial dimensions. It looks very much like the second group is written with a convention in which time indices are raised and spatial indices are lowered. But there's no single, universal convention for this that makes one more meaningful than the other.

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No, there isn't any difference in the physics. However, in this paper are distinguished in an attempt to make things clearer, reserving upper indices for time, and lower indices for spatial dimentions. Still, this is just a choice and you could write everything in subscript as long as you don't confuse time and spatial dependence.

The difference in upper or lower indices matter when you are dealing with tensors, and want to express in which way they transform, lower for covariant and upper for contravariant. In a context of normal three dimensional vectors and working in euclidean space, it would be really unlikely that difference would matter, specially if they're being used on a quantity like temperature that it is just scalar.

In general, the place of indices will not, in most cases, be relevant unless you are dealing with tensors and unless noting the way they transform is relevant (for example, in special/general relativity).

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    $\begingroup$ If the OP's examples, index $l$ represents the time step and $i$ and $j$ represent the 2-dimensional mesh of points in the model. Using a notation like $T_{i,j}$ and $T_{i,l}$ would be confusing since the two pairs of subscripts mean different things. $\endgroup$ – alephzero May 24 at 18:45
  • $\begingroup$ What would T_i,_l refer to? $\endgroup$ – Abinash May 24 at 18:50
  • $\begingroup$ You are totally right, I didn't notice those indices meant spatial/temporal distinction. Still, they are just placed differently with intention of making things clearer, but technically nothing would happen if you decided to use only lowerscripts. In contrast, in tensor notation you would change the meaning of an equation by switching the indices $\endgroup$ – Rafael Rodríguez Velasco May 24 at 18:52
  • $\begingroup$ @Abinash $T_{i,l}$ would be non sense since i indicates one of the spatial dimentions and l indicates time, and not a different spatial dimension. But again, the distinction was meant for the author to make things clearer, it doesn't mean that it would be wrong to write l down, just more confusing. $\endgroup$ – Rafael Rodríguez Velasco May 24 at 18:57

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