Merge pull request #287 from FourierFlows/ncc/enhance-docs-aliasing

Clarifications and better phrasing in Docs/Aliasing

Author: navidcy

docs/src/aliasing.md

@@ -8,9 +8,10 @@ Plots.scalefontsizes(1.25)
 Plots.default(lw=3)
 ```
 
-In pseudospectral methods, often aliasing errors come into play. These errors originate from
-the discrete version of the grid. A grid discretized with ``n_x`` points can only resolve a 
-total of ``n_x`` wavenumbers in Fourier space. 
+In pseudospectral methods, when computing nonlinear terms then aliasing errors come into play. 
+These aliasing errors originate from the discrete nature of the grid and, specifically, with 
+the restriction that a grid discretized with ``n_x`` points can only resolve a total of ``n_x`` 
+wavenumbers in Fourier space. 
 
 On a grid with a total of ``n_x`` wavenumbers, both harmonics ``e^{2\pi i k x / L_x}`` and 
 ``e^{2\pi i (k+n_x) x / L_x}``, with ``k`` an integer, are indistinguishable when evaluated
@@ -41,28 +42,38 @@ savefig("assets/plot4.svg"); nothing # hide
 
 ![](assets/plot4.svg)
 
-The take home message is that on this grid we cannot distinguish harmonics of wavenumbers 4 and 6
-and attempting to represent harmonics with wavenumber 6 on this grid will lead to aliasing errors.
-For example, say that we are solving an equation on this grid and at some point we compute the product 
-``\cos(2x) \cos(4x)``. The result is ``\frac1{2} \cos(2x) + \frac1{2} \cos(6x)``, but on this 
-grid ``\cos(6x)`` is indistinguishable from ``\cos(4x)`` and, therefore, we get an answer 
-which is the sum of ``\frac1{2} \cos(2x) + \frac1{2} \cos(4x)``!
-
-There are two ways to avoid aliasing errors we either *(i)* need to discard some of the wavenumber 
-components in Fourier space before we tranform to physical space, or *(ii)* pad our Fourier 
-represresentation with more wavenumbers that will have zero power. In FourierFlows.jl the former
-is implemented
+The take home message is that on this grid we cannot distinguish harmonics of wavenumbers 4 
+and 6 and attempting to represent harmonics with wavenumber 6 on this grid will lead to aliasing 
+errors. For example, say that we are solving an equation on this grid and at some point we compute 
+the product ``\cos(2x) \cos(4x)``. The result is ``\frac1{2} \cos(2x) + \frac1{2} \cos(6x)``, 
+but on this grid ``\cos(6x)`` is indistinguishable from ``\cos(4x)`` and, therefore, we get an 
+answer which is the sum of ``\frac1{2} \cos(2x) + \frac1{2} \cos(4x)``!
+
+To avoid aliasing errors we either *(i)* discard some of the wavenumber components in Fourier 
+space from our fields before we tranform to physical space, or *(ii)* pad our fields with Fourier 
+components with zero power that correspond to higher wavenumbers than those resolved by the grid 
+before transforming to physical space. This way, the aliasing errors, which will involve the 
+higher wavenumbers, will be either *(i)* zero-ed out or *(ii)* only come about for wavenumbers 
+beyond what our grid can resolve anyway. In FourierFlows.jl, the former dealiasing scheme is 
+implemented.
 
 !!! info "De-aliasing scheme"
     FourierFlows.jl curently implement dealiasing by zeroing out the top-`aliased_fraction` 
     wavenumber components on a `grid`.
 
-How many wavenumber components we need to discard depends on the order of the nonlinearity. For
-quadradic nonlinearities, one would intuitively say that we need to discard the top-1/2 of the 
-wavenumber components. However, Orszag (1972) pointed out that simply only discarding the 
-top-1/3 of wavenumber components is enough. Actally, with Orszag's so-called 2/3-rule for dealiasing, 
-still some aliasing errors occur, but only into wavenumbers that will be zero-ed out next time 
-we dealias.
+The number of wavenumber components that we need to zero-out to be sure the no aliasing errors 
+infiltrate our solution depends on the order of the nonlinearities. For example, for quadratic 
+nonlinearities, one expects that we need to discard the top-1/2 of the wavenumber components. 
+This way, when computing the product of two fields we won't have anything that projects onto
+harmonics with wavenumbers beyond those that our grid is able to resolve and, therefore, no 
+aliasing errors.
+
+The above-mentioned 1/2-rule for dealiasing for quadratic nonlinearities is, however, not the 
+most efficient. Orszag (1972) pointed out that for quadratic nonlirearities, simply only discarding 
+the top-1/3 of wavenumber components is enough to save us from aliasing errors. To be fair, 
+with Orszag's so-called 2/3-rule for dealiasing, still some aliasing errors occur, but those 
+errors only occur to the higher-1/3 wavenumber components that will be zero-ed out next time 
+we dealias our solution anyway.
 
 When constructing a `grid` we can specify the `aliased_fraction` parameter. By default, this is 
 set to ``1/3``, appropriate for quadratic nonlinearities. Then `dealias!(fh, grid)` will zero-out