VlasovMaxwell App: Vlasov-Maxwell equations on a Cartesian grid¶

The VlasovMaxwell app solves the Vlasov equation on a Cartesian grid.

where \(f_s\) the particle distribution function, \(\mathbf{a}_s\) is particle acceleration and \(C[f_s,f_{s'}]\) are collisions. This app uses a version of the discontinuous Galerkin (DG) scheme to discretize the phase-space advection, and Strong-Stability preserving Runge-Kutta time-steppers (SSP-RK) to discretize the time derivative. We use a matrix and quadrature free version of the algorithm described in [Juno2018] which should be consulted for details of the numerics the properties of the discrete system.

For neutral particles, the acceleration can be set to zero. For electromagnetic (or electrostatic) problems, the acceleration is due to the Lorentz force:

The electromagnetic fields can be either specified, or determined from Maxwell or Poisson equations.

Overall structure of app¶

By CDIM we mean configuration space dimension and by VDIM we mean velocity space dimension. NDIM = CDIM+VDIM is the phase-space dimension. Note that we must have VDIM>=CDIM.

The overall structure of the app is as follows

local Vlasov = (require "App.PlasmaOnCartGrid").VlasovMaxwell()

vlasovApp = Vlasov.App {
  -- basic parameters

  -- description of each species: names are arbitrary
  elc = Vlasov.Species {
    -- species parameters
  },

  -- fields (optional, can be omitted for neutral particles)
  field = Vlasov.Field {
    -- field parameters
  },
}
-- run application
vlasovApp:run()

Note that if the app’s run method is not called, the simulation will not be run, but the simulation will be initialized and the initial conditions will be written out to file.

Basic parameters¶

The app takes the following basic parameters. Parameters that have default values can be omitted.

Note that the field object must be called “field”. You can also omit the field object completely. In this case, it will be assumed that you are evolving neutral particles and the acceleration will be set to zero (i.e. \(\mathbf{a}_s = 0\) in the Vlasov equation).

Only one field object (if not omitted) is required. At present, the app supports a EM field evolved with Maxwell equations, or a EM field specified as a time-dependent function.

Species parameters¶

The Vlasov app works with arbitrary number of species. Each species is described using the Vlasov.Species objects. By default every species in the app is evolved. However, species evolution can be turned off by setting the evolve flag to false. Species can be given arbitrary names. As the species names are used to label the output data files, reasonable names should be used.

elc = Vlasov.Species {
  -- species parameters
},

The supported diagnostic moments are, “M0”, “M1i”, “M2ij”, “M2” and “M3i” defined by

In these diagnostics, the index \(i,j\) run over \(1\ldots VDIM\).

Additionally, Gkeyll can calculate the weak moments: bulk velocity “u” and the square of thermal velocity “vtSq”.

The boundary conditions (if not periodic) are specified with the bcx etc. tables. Each table must have exactly two entries, one for BC on the lower edge and one for the upper edge. The supported values are

Note that often “reflection” boundary condition is used to specify a symmetry for particles.

For example, for a 1x simulation, to specify that the left boundary is a reflector, while the right an absorber use:

bcx = { Vlasov.Species.bcReflect, Vlasov.Species.bcAbsorb }

Electromagnetic field parameters¶

The EM field object is used as follows

field = Vlasov.Field {
  -- field parameters
},

Note: When doing an electrostatic problem with no magnetic field, set the hasMagneticField to false. This will choose specialized solvers that are much faster and can lead to significant gain in efficiency.

The boundary conditions (if not periodic) are specified with the bcx etc. tables. Each table must have exactly two entries, one for BC on the lower edge and one for the upper edge. The supported values are

Functional field parameters¶

To peform “test-particle” simulation one can specify a time-dependent electromagnetic field which does not react to particle currents.

externalField = Vlasov.ExternalField {
  -- field parameters
},

App output¶

The app will write distribution function for each species and the EM fields at specified time intervals. Depending on input parameters specified to the species and field block, different number of distribution functions, fields and diagnostics (moments, integrated quantities) will be written.

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

• vlasov_elc_N.bp

• vlasov_ion_N.bp

• vlasov_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.

In addition to the above, optionally diagnostic data may also be written. For example, the moments files are named:

• vlasov_elc_M0_N.bp

• vlasov_ion_M0_N.bp

• vlasov_elc_M1i_N.bp

• vlasov_ion_M1i_N.bp

etc, depending on the entries in the diagnostics table for each species. In addition, integrated moments for each species are written:

• vlasov_elc_intMom_N.bp

This file has the time-dependent “M0”, three contributions of kinetic energy and the “M2” (integrated over configuration space) stored in them.

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.bp

These can be plotted using postgkyl in the usual way.

Examples¶

• Advection in a potential well (Field not evolved)

• Landau damping of Langmuir waves

• Two-stream instability

• Three species electrostatic shock problem. See [Pusztai2018] for full problem description.

• Advection of particles in a constant magnetic field. (Field not evolved)

• Weibel instability in 1x2v. See [Cagas2017] for full problem description.

References¶