Different vertical advection schemes

[simone-silvestri]

The objective of this PR is to implement some stable vertical advection schemes (Upwinding and WENO)
Following http://www.gfdl.noaa.gov/idealized-spectral-models-quickstart, it looks like upwinding with semi-implicit time stepping requires reconstructing advected variables from previous time steps.

An example of 10 days simulation with third order upwinding (left) and second order centered advection (right)

[simone-silvestri]

Difference between upwind third order (top, 3-point stencil) and upwind fifth order (bottom, 5-point stencil)

upwind5-upwind3.mp4

[milankl]

This is awesome!!

The Jablonowski test case (see also #265) can now be run as follows

using SpeedyWeather

# no mountains, no physics
spectral_grid = SpectralGrid(trunc=31,nlev=8)
orography = NoOrography(spectral_grid);

# I believe I deviated slightly here from the Jablonowski initial temperature profile for the stratosphere, will check
initial_conditions = ZonalWind()
model = PrimitiveDryModel(;spectral_grid,orography,initial_conditions,physics=false);
simulation = initialize!(model);

run!(simulation,n_days=10)

which yields

[milankl]

ZonalWind are the Jablonowski initial conditions with perturbation

julia> initial_conditions = ZonalWind()
ZonalWind <: InitialConditions:
 η₀::Float64 = 0.252
 u₀::Float64 = 35.0
 perturb_lat::Float64 = 40.0
 perturb_lon::Float64 = 20.0
 perturb_uₚ::Float64 = 1.0
 perturb_radius::Float64 = 0.1
 ΔT::Float64 = 0.0
 Tmin::Float64 = 200.0
 pressure_on_orography::Bool = false

But I think they use another ΔT to have an increasing temperature in the stratosphere

[simone-silvestri]

The results seem to be roughly the same with all advection schemes in this case (probably, there is not much influence of the vertical advection).
I can try increasing the resolution and seeing if the solution varies.

Is there a way to validate the exactness of the solution (i.e., a reference solution?) Or is there a particularly unstable case we can test to see if there is some stability improvement?
Not convecting cases, though since, even if more stable than centered, upwind advection will not cut it

[simone-silvestri]

Using

spectral_grid = SpectralGrid(trunc=170,nlev=40)

and 30 day time integration

vorticity.mp4

temperature.mp4

[milankl]

Quick speed comparison

[milankl]

@simone-silvestri is there anything you wanted to add here? Have you checked that it's not allocating?

[milankl]

In https://speedyweather.github.io/SpeedyWeather.jl/previews/PR365/primitiveequation/#Vertical-advection (may need some minutes to build) you can see the current state of the documentation. I won't go into details of the reconstruction and what schemes that means, but if you could have a look at that section and feel free to add anything that's important to understand the code! ❤️

[simone-silvestri]

@simone-silvestri is there anything you wanted to add here? Have you checked that it's not allocating?

I think it is ready to go. I will check that it does not allocate but it shouldn't (only tuples)
I ll also take a look at the documentation and fill it accordingly

[simone-silvestri]

This seems to be ready, should I merge and look at the docs in your new PR? Should I add some tests before?