Dealiasing parameter for quad,cub truncation

[milankl]

@hottad this introduces dealiasing as a parameter (1=linear, 2=quadratic, 3=cubic) but you can also chose anything in between and it chooses the right grid size given the truncation trunc (and dealiasing). I therefore removed anything that related to the spectral resolution from RingGrids and SpeedyTransforms now has the following methods

const DEFAULT_DEALIASING = 2.0    # quadratic truncation default

function RingGrids.get_nlat_half(   trunc::Integer,
                                    dealiasing::Real=DEFAULT_DEALIASING)
                    
    return roundup_fft(ceil(Int,((1+dealiasing)*trunc+1)/4))
end

function get_truncation(nlat_half::Integer,
                        dealiasing::Real=DEFAULT_DEALIASING)

    return floor(Int,(4nlat_half-1)/(1+dealiasing))
end

to match spectral and grid-point resolution as desired.

[milankl]

OctaHEALPixGrid is 1.6x faster than FullGaussianGrid when both are used at quadratic truncation:

There are visible differences, does it matter? I don't know, we will see. The OctahedralGaussianGrid is still exact at quadratic so no visible differences but considerably slower than OctaHEALPix

[milankl]

For some reason this still throws an error in the first spectral transform

julia> run_speedy(PrimitiveDryCore,Grid=HEALPixGrid,dealiasing=2)
ERROR: BoundsError
Stacktrace:
 [1] gridded!(map::HEALPixGrid{Float64}, alms::LowerTriangularMatrix{ComplexF64}, S::SpectralTransform{Float64}; unscale_coslat::Bool)
   @ Main.SpeedyWeather.SpeedyTransforms ~/git/SpeedyWeather.jl/src/SpeedyTransforms/spectral_transform.jl:355
 [2] gridded!
   @ ~/git/SpeedyWeather.jl/src/SpeedyTransforms/spectral_transform.jl:340 [inlined]
 [3] initialize_orography!(orog::Main.SpeedyWeather.Orography{Float64, HEALPixGrid{Float64}}, #unused#::EarthOrography, P::Parameters{PrimitiveDryCore}, S::SpectralTransform{Float64}, G::Geometry{Float64})

[milankl]

The last commit resolves the issue that HEALPix errored with quadratic truncation. However, there are clearly issues with the HEALPixGrid when using quadratic truncation: This is T63 with

• (top) O48, OctahedralGaussian, 96 rings, 208 points at Equator
• (middle) P48, OctaHEALPix, 95 rings, 192 points at Equator
• (bottom) H48, HEALPix, 95 rings, 96 points at Equator

Note all the aliasing with HEALPix, which however goes basically away with OctaHEALPix! 🎉

[hottad]

Great work! Thank you, Milan!

Meridional quadrature in HEALPixGrid and OctaHEALPixGrid are inexact by design, so it is natural that you see visibly different results with HEALPixGrid at a relatively low resolution of T63.
It will be interesting to compare at a much higher resolution since the inexact quadrature in HEALPixGrid and OctaHEALPixGrid becomes more and more accurate with increased resolution. From experience with ectrans (IFS's spectral transform code), the "inexactness" of the quadrature becomes irrelevant at around T479 resolution, in the sense that single-precision transforms on these grids become as accurate as single-precision transforms on the tried-and-truth Gaussian or Clenshaw-Curtis grids.

I found it surprising that T63 on H48 did not blow up (albeit with easily recognizable aliasing noises)!
With T63, purely zonal wave can have zonal wavenumber of 63, which requires (63+1)*2=128 zonal grid points along the Equator even to avoid linear aliasing, while H48 only has 96 points along the tropical belt.

It's great that this problem goes away with OctaHEALPix. It is exactly for this reason that I suggested OctaHEALPix, so it is just consistent with the theory, but it is always nice to see that an implementation actually works as expected from theory.

Thank you! and Congratulations!