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docs/src/aliasing.md

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@@ -58,26 +58,26 @@ beyond what our grid can resolve anyway. In FourierFlows.jl, the former dealiasi
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implemented.
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!!! info "De-aliasing scheme"
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FourierFlows.jl curently implements dealiasing by zeroing out the top-`aliased_fraction`
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FourierFlows.jl curently implements dealiasing by zeroing out the highest-`aliased_fraction`
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wavenumber components on a `grid`.
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The number of wavenumber components that we need to zero-out to be sure the no aliasing errors
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infiltrate our solution depends on the order of the nonlinearities. For example, for quadratic
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nonlinearities, one expects that we need to discard the top-1/2 of the wavenumber components.
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nonlinearities, one expects that we need to discard the highest-1/2 of the wavenumber components.
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This way, when computing the product of two fields we won't have anything that projects onto
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harmonics with wavenumbers beyond those that our grid is able to resolve and, therefore, no
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aliasing errors.
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The above-mentioned 1/2-rule for dealiasing for quadratic nonlinearities is, however, not the
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most efficient. Orszag (1972) pointed out that for quadratic nonlirearities, simply only discarding
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the top-1/3 of wavenumber components is enough to save us from aliasing errors. To be fair,
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the highest-1/3 of wavenumber components is enough to save us from aliasing errors. To be fair,
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with Orszag's so-called 2/3-rule for dealiasing, still some aliasing errors occur, but those
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errors only occur to the higher-1/3 wavenumber components that will be zero-ed out next time
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we dealias our solution anyway.
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When constructing a `grid` we can specify the `aliased_fraction` parameter. By default, this is
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set to ``1/3``, appropriate for quadratic nonlinearities. Then `dealias!(fh, grid)` will zero-out
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the top-`aliased_fraction` wavenumber components of `fh`.
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the highest-`aliased_fraction` wavenumber components of `fh`.
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If we construct a grid with `aliased_fraction=0`, e.g.,
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