Collision models in Gkeyll

In Gkeyll we currently have two different collision operators, one is the Bhatnagar–Gross–Krook (BGK) collision operator and the other is the Dougherty operator. We referred to the latter as the LBO for the legacy of Lenard-Bernstein.

BGK collisions

The BGK operator [Gross1956] is for the effect of collisions on species \(f_s\) is

\[\left(\frac{\partial f_s}{\partial t}\right)_c = \sum_r\nu_{sr} \left(f_s - f_{Msr}\right)\]

where the sum is over all the species. The distribution functon \(f_{Msr}\) is the Maxwellian

\[f_{Msr} = \frac{n_s}{\left(2\pi v_{tsr}^2\right)^{d_v/2}} \exp\left[-\frac{\left(\mathbf{v}-\mathbf{u}_{sr}\right)^2}{2v_{tsr}^2}\right]\]

with the primitive moments \(\mathbf{u}_{sr}\) and \(v_{tsr}^2\) properly defined to preserve some properties (such as conservation), and \(d_v\) is the number of velocity-space dimensions. For self-species collisions \(\mathbf{u}_{sr}=\mathbf{u}_s\) and \(v_{tsr}^2=v_{ts}^2\). For multi-species collisions we follow an approach similar to [Greene1973] and define the cross-species primitive moments as

\[\begin{split}\mathbf{u}_{sr} &= \mathbf{u}_s - \frac{\alpha_{E}}{2} \frac{m_s+m_r}{m_sn_{s}\nu_{sr}}\left(\mathbf{u}_s-\mathbf{u}_r\right) \\ v_{tsr}^2 &= v_{ts}^2 - \frac{1}{d_v}\frac{\alpha_E}{m_sn_{s}\nu_{sr}} \left[d_v\left(m_sv_{ts}^2-m_rv_{tr}^2\right)-m_r\left(\mathbf{u}_s-\mathbf{u}_r\right)^2 +4\frac{\alpha_E}{m_sn_{s}\nu_{sr}}\left(m_s+m_r\right)^2\left(\mathbf{u}_s-\mathbf{u}_r\right)^2\right]\end{split}\]

but contrary to Greene’s definition of \(\alpha_E\), we currently use in Gkeyll the following expression

\[\alpha_E = m_sn_{s}\nu_{sr}\delta_s\frac{1+\beta}{m_s+m_r}.\]

Little guidance is provided by Greene as to how to choose \(\beta\), although it seems clear that \(-1<\beta\). In Gkeyll the default value is \(\beta=0\), but the user can specify it in the input file (explained below). We have introduced the additional quantity \(\delta_s\) (which Greene indirectly assumed to equal 1) defined as

\[\delta_s = \frac{2m_sn_s\nu_{sr}}{m_sn_s\nu_{sr}+m_rn_r\nu_{rs}}\]

The BGK operator can be used with both the Vlasov-Maxwell solver and the gyrokinetic solver.

Dougherty collisions

The Doughery (LBO) model for collisions [Dougherty1964] in Gkeyll is given by

\[\left(\frac{\partial f_s}{\partial t}\right)_c = \sum_r\nu_{sr} \frac{\partial}{\partial\mathbf{v}}\cdot\left[\left(\mathbf{v}-\mathbf{u}_{sr}\right)f_s +v_{tsr}^2\frac{\partial f_s}{\partial\mathbf{v}}\right].\]

In this case we compute the cross-primitive moments by a process analogous to Greene’s with the BGK operator, yielding the following formulas for the cross flow velocity and thermal speed:

\[\begin{split}\mathbf{u}_{sr} &= \mathbf{u}_s + \frac{\alpha_{E}}{2} \frac{m_s+m_r}{m_sn_{s}\nu_{sr}}\left(\mathbf{u}_r-\mathbf{u}_s\right) \\ v_{tsr}^2 &= v_{ts}^2+\frac{\alpha_{E}}{2}\frac{m_s+m_r}{m_sn_{s}\nu_{sr}} \frac{1}{1+\frac{m_s}{m_r}}\left[v_{tr}^2-\frac{m_s}{m_r}v_{ts}^2 +\frac{1}{d_v}\left(\mathbf{u}_s-\mathbf{u}_r\right)^2\right]\end{split}\]

with \(\alpha_E\) defined in the BGK section above. The LBO used by the gyrokinetic solver is

\[\left(\frac{\partial f_s}{\partial t}\right)_c = \sum_r\nu_{sr}\left\lbrace \frac{\partial}{\partial v_{\parallel}}\left[\left(v_\parallel-u_{\parallel sr}\right)f_s +v_{tsr}^2\frac{\partial f_s}{\partial v_\parallel}\right]+\frac{\partial}{\partial\mu} \left[2\mu f_s+2\frac{m_sv_{tsr}^2}{B}\mu\frac{\partial f_s}{\partial\mu}\right]\right\rbrace\]

Collisions in Gkeyll input files

Users can specify collisions in input files with either constant collisionality or with spatially varying, time-evolving collisionality.

Constant collisionality

An example of adding LBO collisions (for BGK collisions simply replace LBOcollisions with BGKCollisions) to a species named ‘elc’ is

elc = Plasma.Species {
   charge = q_e, mass = m_e,
   -- Velocity space grid.
   lower = {-6.0*vt_e},
   upper = { 6.0*vt_e},
   cells = {32},
   -- Initial conditions.
   init = Plasma.MaxwellianProjection{
      density = function (t, zn)
         local x, vpar = zn[1], zn[2]
         return n_e
      end,
      driftSpeed = function (t, zn)
         local x, vpar = zn[1], zn[2]
         return {u_e}
      end,
      temperature = function (t, zn)
         local x, vpar = zn[1], zn[2]
         return m_e*(vt_e^2)
      end,
   },
   evolve = true,
   -- Collisions.
   coll = Plasma.LBOCollisions {
      collideWith = { "elc" },
      frequencies = { nu_ee },
   },
},

If there were another species, say one named ‘ion’, this ‘elc’ species could be made to collide with ‘ion’ by adding ‘ion’ to the collideWidth table:

coll = Plasma.LBOCollisions {
   collideWith = { "elc", "ion" },
   frequencies = { nu_ee, nu_ei },
},

The constant collision frequencies nu_ee and nu_ei need to be previously computed/specified in the input file. The user can specify the value of \(\beta\) in the above formulas for the cross-species primitive moments (\(\mathbf{u}_{sr}\) and \(v_{tsr}^2\)) by specifying the variable betaGreene in the collisions table. If the user does not specify it, it is assumeed that betaGreene=0.0.

In some cases the user may be interested in colliding species ‘elc’ with species ‘ion’, but not collide species ‘ion’ with species ‘elc’. Gkeyll supports this combination, but since the formulas for cross-species primitive moments involve both \(\nu_{ei}\) and \(\nu_{ie}\), the code will default to assuming \(\nu_{ie}=m_e\nu_{ei}/m_i\). Note however that this scenario is not energy conserving: for exact energy conservation, one must include the effect of binary collisions on both species.

It is also possible to specify both LBO and BGK collisions between different binary pairs in a single input file. For example, if there are three species ‘elc’, ‘ion’ and ‘neut’, the ‘elc’ species could be made collide with both ‘ion’ and ‘neut’ as follows:

cColl = Plasma.LBOCollisions {
   collideWith = { "elc", "ion" },
   frequencies = { nu_ee, nu_ei },
},
nColl = Plasma.BGKCollisions {
   collideWith = { "neut" },
   frequencies = { nu_en },
},

If no collisionality is specified in the input file, it is assumed that the user desires Gkeyll to build a spatially-varying collisionality from scratch using a Spitzer-like formula for \(\nu_{sr}\) (explained below).

Spatially varying collisionality

The simplest way to run with spatially varying collisionality is to not specify the table frequencies. In this case the code computes \(\nu_{sr}\) according to

\[\nu_{sr} = \nu_{\mathrm{frac}}\frac{n_r}{m_s}\left(\frac{1}{m_s}+\frac{1}{m_r}\right) \frac{q_s^2q_r^2\log\Lambda_{sr}}{3(2\pi)^{3/2}\epsilon_0^2} \frac{1}{\left(v_{ts}^2+v_{tr}^2\right)^{3/2}}\]

where \(\nu_{\mathrm{frac}}\) is a scaling factor, the Coulomb logarithm is defined as

\[\log\Lambda_{sr} = \ln\left\lbrace\left(\sum_\alpha\frac{\omega_{p\alpha}^2+\omega_{c\alpha}^2} {\frac{T_\alpha}{m_\alpha}+3\frac{T_s}{m_s}}\right)^{-1/2} \left[\max\left(\frac{|q_sq_r|}{4\pi\epsilon_0m_{sr}u^2},\frac{\hbar}{2e^{1/2}m_{sr}u}\right)\right]^{-1}\right\rbrace\]

and the \(\alpha\)-sum is over all the species. For Vlasov-Maxwell simulations we do not add the correction due to gyromotion (\(\omega_{c\alpha}=0\) here). The relative velocity here is computed as \(u=3v_{tr}^2+3v_{ts}^2\), the reduced mass is \(m_{sr} = m_sm_r/\left(m_s+m_r\right)\), and \(\omega_{p\alpha}\) is the plasma frequency computed with the density and mass of species \(\alpha\). Simpler formulas for the Coulomb logarithm can be easily generated by developers if necessary.

The formulas above assume all the plasma quantities and universal constants are in SI units. The user can provide a different value for these variables by passing them to the collisions table in the input files, as shown here:

coll = Plasma.LBOCollisions {
   collideWith = { "elc", "ion" },
   epsilon0    = 1.0,    -- Vacuum permitivity.
   elemCharge  = 1.0,    -- Elementary charge value.
   hBar        = 1.0,    -- Planck's constant h/2pi.
},

Additionally the user can pass the scaling factor \(\nu_{\mathrm{frac}}\) by specifying nuFrac in the collisions table.

Another way to use a spatially varying collisionality is to pass a reference collisionality normalized to some values of density and temperature. For example if the input file specifies the normalized collisionality \(\nu_{srN}=\nu_{sr0}\left(T_{s0}/m_s+T_{r0}/m_r\right)^{3/2}/n_{r0}\) through normNu

coll = Plasma.LBOCollisions {
   collideWith = { "elc" },
   normNu      = { nu_ee*(2*T_e0/m_e)^(3/2)/n_e0 }
},

then in each time step the collisions will be applied with the following collisionality

\[\nu_{sr}(x) = \nu_{\mathrm{frac}}\nu_{srN} \frac{n_r(x,t)}{\left(v_{ts}^2(x,t)+v_{tr}^2(x,t)\right)^{3/2}}.\]

Currently these options lead to a spatially varying, cell-wise constant collisionality. We will be adding support for variation of the collisionality within a cell in the future.

Comments on stability

The are known issues with the implementation of the collision operators in Gkeyll. One of them, for example, is that we do not have a positivy preseving algorithm for the LBO. Positivity issues are often accompanied by large flows or negative temperatures and/or densities. For this reason we have taken three precautions:

  1. Calculation of primitive moments \(\mathbf{u}_{sr}\) and \(v_{tsr}^2\) is carried out using cell-average values if the number density is non-positive at one of the corners of that cell.
  2. The collision term is turned off locally if the flow velocity \(\mathbf{u}_{sr}\) is greater than the velocity limits of the domain, or if \(v_{tsr}^2\) is negative.
  3. The collision frequency \(\nu_{sr}\) is locally set to zero if the cell-average values of \(n_r\) or \(v_{tsr}^2\) are negative.

We track the number of cells in which precaution 2 is used, and for stable simulations this is typically small (a few percent or less). Further discussion of why these precautions are necessary appears in [Hakim2019].

References

[Gross1956]Gross, E. P. & Krook, M. (1956) Model for collision precesses in gases: small-amplitude oscillations of charged two-component systems. Physical Review, 102(3), 593–604.
[Greene1973]Greene, J. M. (1973). Improved Bhatnagar-Gross-Krook model of electron-ion collisions. Physics of Fluids, 16(11), 2022–2023.
[Dougherty1964]Dougherty, J. P. (1964). Model Fokker-Planck Equation for a Plasma and Its Solution. Physics of Fluids, 7(11), 1788–1799.
[Hakim2019]Hakim, A., et. al. (2019). Conservative Discontinuous Galerkin Schemes for Nonlinear Fokker-Planck Collision Operators. Submitted to the Journal of Computational Physics, arXiv:1903.08062.