Moments App: Multifluid-moment-Maxwell model¶

Summary of model equations¶

The Moment app solves high-moment multifluid equations on a Cartesian grid, coupled to Maxwell’s equations through the Lorentz force.

• Each fluid could be either five-moment, where the plasma pressure is assumed to a scalar (see [Hakim+2006]), or ten-moment, where the full anisotropic and nongyrotropic plasma pressure tensors (see [Hakim2008]) are evolved.

• The hyperbolic system is solved in converative forms of the five-moment (Euler) equations, and/or the ten-moment equations, and the perfectly hyperbolic Maxwell’s equations.

• The sources that couples the plasma species momenta and electromagnetic fields are described here and more comprehensively in [Wang+2020].

This App solves the hyperbolic and source parts parts of the coupled system separately and apply high accuracy schemes on both.

• Ignoring sources, the homogeneous equations of the fluid moments and electromagneic fields are solved separately using a high-resolution wave-propagation finite-volume method described in [Hakim+2006]. The main time-step size constraint comes from the speed of light.

• The sources are evolved using a locally implicit, time-centered solver to step over the constraining scales like the Debye length and plasma frequency, etc. See Handling two-fluid five-moment and ten-moment source terms or [Wang+2020] for more details.

• Additonial sources can be added if needed, for example, for the ten-moment model, we may apply the closure to relax the pressure tensor towards a scalar pressure (see [Wang+2015]).

• We then apply an Strang-type operator-splitting sequence to combine the hyperbolic and source parts to achive second-order accuracy in time:

  \[\exp\left(\mathcal{L}_{S}\Delta t/2\right)\exp\left(\mathcal{L}_{H}\Delta t\right)\exp\left(\mathcal{L}_{S}\Delta t/2\right).\]

  Here, we represent the homogeneous update schematically as the operator \(\exp\left(\mathcal{L}_{H}\Delta t\right)\) and the source update as \(\exp\left(\mathcal{L}_{S}\Delta t\right)\).

Overall structure of the Moments app¶

--------------
-- Preamble --
--------------
-- The Moments app wraps fluid and field objects, and tells the
-- the program how to evolve and couple them
local Moments = require("App.PlasmaOnCartGrid").Moments()
local TenMoment = require "Eq.TenMoment" -- TenMoment or Euler

-- Create the app
momentApp = Moments.App {
  ------------
  -- COMMON --
  ------------
  -- basic parameters, e.g., time step, grid, domain decomposition

  -- Description of each species: names are arbitrary
  electron = Moments.Species {
    -- species parameters, equations, and boundary conditions
  },

  -- Repeat to add more species
  hydrogen = Moments.Species { ... },
  oxygen = Moments.Species { ... },

  -- EM fields (optional, can be omitted for neutral fluids)
  field = Moments.Field {
    -- EM field parameters, equations, and boundary conditions
  },

  -- Basic source that couple the fluids and EM fields
  emSource = Moments.CollisionlessEmSource {
     -- names of the species to be coupled
     species = {"electron", "hydorgen", "oxygen"},
     -- other specifications
  },

  -- Additional sources if needed
  elc10mSource = Moments.TenMomentRelaxSource {
     species = {"elctron"},
     -- other specifications
  },
}

-- run the app
momentApp:run()

Examples¶

• Five-moment modeling of the GEM challenge magnetic reconnection problem.

• Ten-moment modeling of the GEM challenge magnetic reconnection problem. This simulation uses a simplified closure appropriate for this problem.

Basic parameters¶

Species parameters¶

Electromagnetic field parameters¶

App output¶

The app will write snapshots of moments for each species and the EM fields at specified time intervals. Diagnostics like integrated fluid moments and field energy are recorded for each time-step and written in one file for each species/field object.

The output format is ADIOS BP files. Say your input file is called “5m.lua” and your species are called “elc” and “ion”. Then, over specified time invertals the app will write out the following files:

• 5m_elc_N.bp

• 5m_ion_N.bp

• 5m_field_N.bp

Where N is the frame number (frame 0 is the initial conditions). Note that if a species or the field is not evolved, then only initial conditions will be written unless the forceWrite option is set to true.

In addition, integrated moments for each species are written:

• vlasov_elc_intMom_N.bp

For the field, the electromagnetic energy components \(E_x^2\), \(E_y^2\), \(E_z^2\), \(B_x^2\), \(B_y^2\), and \(B_z^2\) (integrated over configuration space) are stored in the file:

• vlasov_fieldEnergy_N.bp

These can be plotted using postgkyl in the usual way.

References¶