mitmath / 18337sp2023-vincent_fan___-adaptive-hermite-refinement

adaptive-hermite-refinement

Open Questions (assignee)

1. Do higher level moments have more spatial dimensions? (Alex) No, but in general yes. One of the convinent things this set of eqs does is decouple perpendicular and parallel velocity in the equations. So the hermite expansion is really in parallel velocity. In principle there could also be a hermite expansion in different velocity directions, but we dont need to worry about this thanks to the reduced equations used here. TL;DR is that all moments will be funcitons of x,y,(z),time only.
2. Do we save intermediate moment values or only care about the final result? (Alex) Ans: If intermediate means at different timesteps, then yes, probably as a data file every n timesteps, controlled by user. (should not just be stored as giant array, that would be ridiculous)
3. Where in kx/ky space do we even refine?
   • you refine in x and y. You only transform into spectral space to compute the derivatives, and then you transform back.

Useful Links

Google Sheet: https://docs.google.com/spreadsheets/d/11NXlTH3tLCnKrHIvjqpDjWG_UwNBUd8l5Uyi1HydtSo/edit?usp=sharing

Governing Equations

TODO: Vincent:

• Hermite moment equation (7-12 in the paper)
• Spectral version
• Discretized version
  • just discretize the spatial dimension to compute the time derivative. It seems straightforward to take the time derivative and numerically integrate that with some scheme (Runge-Kutta or something else)

Pseudocode

# setup grid
nMoments = Int[kx][ky] # how many moments at each point
moments, tmp_moments = Array{Moment}[kx][ky] # TODO moment dimensionality

# initialize, potentially via Fourier transform of input

# time-integration
for t in 1..T begin
  # don't refine every time ideally
  if t % REFINE_INTERVAL == 0 begin
    # for each point in space, check condition and update nMoments
    for kx in 1..KX begin
      for ky in 1..KY begin
        nMoments[kx][ky] = refineMoments(nMoments[ky][kx], moments[ky][kx])
      end
    end
  end
   
  for kx in 1..KX begin
    for ky in 1..KY begin
      # TODO do the first two moments
      for m in 3..nMoments[kx][ky]
        tmp_moments[kx][ky][m] = f(moments) # TODO ...
      end
    end
  end
  
  moments = tmp_moments
  # plot/output?
end