# 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.

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
},

elc10mSource = Moments.TenMomentRelaxSource {
species = {"elctron"},
-- other specifications
},
}

-- run the app
momentApp:run()


## Basic parameters¶

Basic Parameters for PlasmaOnCartGrid.Moments
Parameter Description Default
logToFile If set to true, log messages are written to log file true
tEnd End time of simulation
suggestedDt Initial suggested time-step. Adjusted as simulation progresses. tEnd/nFrame
nFrame Number of frames of data to write. Initial conditions are always written. For more fine-grained control over species and field output, see below.
lower CDIM length table with lower-left configuration space coordinates
upper CDIM length table with upper-right configuration space coordinates
cells CDIM length table with number of configuration space cells
cfl CFL number to use. This parameter should be avoided and cflFrac used instead. Determined from cflFrac
cflFrac Fraction (usually 1.0) to multiply CFL determined time-step. Determined from timeStepper
maximumDt Hard limit of time step size. tEnd-tStart
timeStepper The multifluid-Maxwell model currently only supports the dimensional- splitting finite-volume method, i.e., "fvDimSplit". "fvDimSplit"
decompCuts CDIM length table with number of processors to use in each configuration space direction. { }
useShared Set to true to use shared memory. false
periodicDirs Periodic directions. Note: X is 1, Y is 2 and Z is 3. E.g., {2} sets the Y direction to be periodic. { }

Note

• In general, you should not specify cfl or cflFrac, unless either doing tests or explicitly controlling the time-step. The app will determine the time-step automatically.
• When useShared=true the decompCuts must specify the number of nodes and not number of processors. That is, the total number of processors will be determined from decompCuts and the number of threads per node.

## Species parameters¶

Parameters for Moments.Species
Parameter Description Default
charge Species charge (ignored for neutral particles)
mass Species mass (ignored for neutral particles)
equation The type of default moment equation for this species, e.g., Euler {gasGamma=5/3}, equation = TenMoment {}. If domain invariance is violated, i.e., negative density/pressure occurs, the step is retaken using the equationInv method that is supposed to guarantee positivity but is more diffusive.
equationInv Backup equation that guarantees positivity in case it is violated when the default equation is used. Examples are: equationInv = Euler { gasGamma = gasGamma, numericalFlux = 'lax' }, equationInv = TenMoment { numericalFlux = "lax" }.
init Function with signature function(t,xn) that initializes the species moments. This function return n values, where n is the number of moments for this species.
bcx Length two table with BCs in X direction. See details on BCs below. { }
bcy Length two table with BCs in Y direction. Only needed if CDIM>1 { }
bcz Length two table with BCs in Z direction. Only needed if CDIM>2 { }
evolve If set to false the moments are not evolved in the hyperbolic part, but could be modified in the source updater. In this case, by default only initial conditions for this species will be written to file. To force writing to file as usual, set the forceWrite option to true. true
forceWrite If set to true the moments are written to file even if evolve is set to false. false

## Electromagnetic field parameters¶

Parameters for Moments.Field derived from App.Field.MaxwellField
Parameter Description Default
nFrame These many field outputs will be written during simulation. If not specified, top-level nFrame parameter will be used nFrame from top-level
epsilon0 Vacuum permittivity ($$\epsilon_0$$)
mu0 Vacuum permeability ($$\mu_0$$)
mgnErrorSpeedFactor Factor specifying speed for magnetic field divergence error correction 0.0
elcErrorSpeedFactor Factor specifying speed for electric field divergence error correction 0.0
init Function with signature function(t,xn) that initializes the field. This function must return 6 values arranged as $$E_x, E_y, E_z, B_x, B_y, B_z$$ at $$t=0$$ at xn, which is a CDIM vector.
bcx Length two table with BCs in X direction. See details on BCs below. { }
bcy Length two table with BCs in Y direction. Only needed if CDIM>1 { }
bcz Length two table with BCs in Z direction. Only needed if CDIM>2 { }
evolve If set to false the field is not evolved. In this case, only initial conditions will be written to file. true
forceWrite If set to true the moments are written to file even if evolve is set to false. false

## 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¶

 [Hakim+2006] (1, 2) Hakim, A., Loverich, J., & Shumlak, U. (2006). A high resolution wave propagation scheme for ideal Two-Fluid plasma equations. Journal of Computational Physics, 219, 418–442. https://doi.org/10.1016/j.jcp.2006.03.036
 [Hakim2008] Hakim, A. H. (2008). Extended MHD modeling with the ten-moment equations. Journal of Fusion Energy, 27, 36–43. https://doi.org/10.1007/s10894-007-9116-z
 [Wang+2020] (1, 2) Wang, L., Hakim, A. H., Ng, J., Dong, C., & Germaschewski, K. (2020). Exact and locally implicit source term solvers for multifluid-Maxwell systems. Journal of Computational Physics. https://doi.org/10.1016/j.jcp.2020.109510
 [Wang+2015] Wang, L., Hakim, A. H. A. H., Bhattacharjee, A., & Germaschewski, K. (2015). Comparison of multi-fluid moment models with particle-in-cell simulations of collisionless magnetic reconnection. Physics of Plasmas, 22(1), 012108. https://doi.org/10.1063/1.4906063