FAQ

1. How do I incorporate scalar or vector field input data where you have a function of the embedded coordinates?

We can take a look at the Brusselator page which sets the values of each point on its mesh to a value as determined by some function. This can be done in a similar manner in both 1D and 2D.

The Brusselator also demonstrates, with the variable F, how one can go about changing the function by which these values are set over time.

2. How do I incorporate input data from a file with linear interpolation?

The Grigoriev Ice Cap model has a section where after the initial data is loaded from a TIF and the data is interpolated so that it may fit over a discrete mesh of our choosing. You may view that here.

3. How do I set boundary conditions like fixed value, no-flux, and no-slip?

Boundary conditions can be set by using "collages", which can take two variables among two different Decapodes and apply a function on the first.

A general workflow would be to have the first Decapode encode the physics and have a second Decapode encode values for the boundary conditions. They can be related by a function that will mask the first variable and replace the desired values with those of the second's. An example of applying fixed boundary conditions would be in the Brusselator page.

A similar workflow can be used for "no-flux" and "no-slip" conditions by fixing the value of the appropriate variable to be 0.

4. How do I plot derived quantities?

Plotting in DECAPODES is commonly done with the Makie package in Julia. Makie allows for creating both still images, which are useful for visualizing the mesh itself and initial/final conditions, and videos, which can capture the full simulation from start to end.

• For 1D visualization

• For 2D visualization

• For 3D visualization

5. How do I add artificial diffusion for 0- or 1-forms?

Without viscosity - i.e. when $μ = 0$ - the incompressible (inviscid) Navier-Stokes equations can be formulated like so:

eq11_inviscid_poisson = @decapode begin
  d𝐮::DualForm2
  𝐮::DualForm1
  ψ::Form0

  ψ == Δ⁻¹(⋆(d𝐮))
  𝐮 == ⋆(d(ψ))

  ∂ₜ(d𝐮) ==  (-1) * ∘(♭♯, ⋆₁, d̃₁)(∧ᵈᵖ₁₀(𝐮, ⋆(d𝐮)))
end

Adding a viscosity term can be accomplished by simply added the appropriate term, and declaring the $μ$ constant:

eq11_viscid_poisson = @decapode begin
  d𝐮::DualForm2
  𝐮::DualForm1
  ψ::Form0
  μ::Constant

  ψ == Δ⁻¹(⋆(d𝐮))
  𝐮 == ⋆(d(ψ))

  ∂ₜ(d𝐮) ==  μ * ∘(⋆, d, ⋆, d)(d𝐮) + (-1) * ∘(♭♯, ⋆₁, d̃₁)(∧ᵈᵖ₁₀(𝐮, ⋆(d𝐮)))
end

More demonstrations on how to iterate between formulations of the same physics (the incompressible Navier-Stokes equations) is available in further detail on the Vortices docs page and in the script available there.

6. How do I use multigrid methods?

To use multigrid methods in the Laplacian solver, you need to create a PrimalGeometricMapSeries that will take a coarse mesh and apply a subdivision method to it some number of times. After that, just use this result as you would a regular mesh for simulation.

s = triangulated_grid(100,100,10,10,Point3D)

# Binary subdivide 4 times
series = PrimalGeometricMapSeries(s, binary_subdivision_map, 4);

# Retrieve highest resolution mesh
sd = finest_mesh(series)

  ...

f_mg = sim_mg(series, generate);

7. What are general workflows for DECAPODES?

A common workflow is to iterate through multiple different models as is done in the Vorticity Model page. A formulation is first done with a direct vorticity formulation but a quick run finds that this setup is unstable. A second formulation introduces a Laplacian solve which produces nice results.

Similar workflows may retain the same model but may iterate on the types of meshes/initial conditions used. An excellent example of this is found in the Glacial Flow page where the model is first run in a 1D setting and then quickly promoted to both 2D and 3D. This allows either running some dynamics in a more complicated setting, as just discussed, or allows for simplifying higher dimensional models by some sort of symmetry.

Troubleshooting

Debugging Generated Simulations

The Decapodes workflow requires generating the simulation function by evaluating the output of gensim. However this simulation code can be saved to a file and edited directly, allowing for full control over the simulation. By this same token, reading the simulation code from a file allows for line numbers to be associated to buggy lines!

open("filename.jl" "w") do f
    write(f, string(gensim(your_decapode)))
end
simulator = include("filename.jl")