Tuesday, June 14, 2005

What's in a Name?

Mathematicians, being exact scientists, use many symbols to denote entities, properties and whatever more they are working with. The problem comes with the size of the alphabet: it is way too small. I can sit down with a numerical colleague, walk through the 26-letter Latin alphabet and we'll probably come up with the same meaning for most of the letters (a-c are constants, d is the differential operator, e is the base of the natural logarithm, [..], j and k are used for subscripting, etc.) Doing the same with a physisist will have different results (c is the speed of light, e is energy, k is a wave number). This is not so bad though, the contexts are different enough to distinguish between them. Besides, we can use capital letters, add the greek alphabet, and use calligraphic notation as well.
But take e for example, is it total energy (or is that E?), internal energy (or is that ε?), or just energy per unit mass (or is that e/ρ). Only within subgroups, scientists may have conventions for this. As long as you mention in the surrounding text what your symbols mean, there is no problem. I like being consistent though, at least throughout my own work. So, being in the start of my PhD, I find this the right time to decide on my own naming conventions. Hence, I find myself doubting about smallcaps e's versus capital E's, counting their occurrences in other's publications. What a job; having time for this! (OK, making time for this)

Names of (numerical) methods also lead to confusion; the main cause here is not the lack of symbols, but the overwhelming amount of methods. It took me some time to figure out the subtle differences between Lax-Friedrichs (LF) and local LF, MUSCL-type interpolation and MUSCL-type fluxes, slope limiters and flux limiters, flux averaging and flux differencing and still have a lot more to figure out.
Unofficial self-invented names may sometimes even be incorrect. When restricting a two-dimensional model to one spatial dimension, everyone agrees on calling this `1.5D'. Doing the same, starting with a three-dimensional model, Tóth and Li incorrectly call this `2.5D'. I agree with Keppens on calling this `1.75D'. After all, there is still only one spatial direction considered.

Finally, most words have many meanings. I am working on adaptive techniques for solving partial differential equations; I use adaptive meshes. But often, these are also called adaptive grids. Or people speak of grid points, gridding, grid generation.
What's wrong with this? Basically nothing, except that the last few years grid computing has become an enormous hype, with publications and conferences all over. And `their' grid is completely different from `our' grid. So, I hereby call on all of my fellow adaptive mathematicians: let's forget about gridding, use meshing, meshes, mesh points (don't forget the whitespace), meshed up, etc.

To conclude: symbol conventions will probably remain different for ever, are not problematic, just annoying for some. And naming conventions can sometimes easily be decided on (if one of them is incorrect). In other cases, it depends on whether people are willing to give up the terms that they have been using for years. The latter is wishful thinking, I guess.