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Collision Operators

In GUERNICA, all coupling across velocity space enters through collision operators. Configuration-space transport is handled independently for each discrete velocity, while collisions introduce either local or global interactions in velocity space.

This page describes the numerical treatment of each collision operator, combining the physical models introduced in the theory with the discrete velocity formulation described in the numerics section.

Collision operator decomposition

The total collision operator acting on the neutral distribution is written as

\[ C[f_n] = C_{\mathrm{iz}}[f_n] + C_{\mathrm{cx}}[f_n] + C_{\mathrm{bgk}}[f_n], \]

corresponding to electron-impact ionization, charge exchange with ions, and neutral–neutral elastic collisions.

In the discrete velocity method (DVM), these operators act on the set of values

\[ \{ f_{n,j}(\mathbf{x},t) \}_{j=1}^{N_v}, \]

and differ primarily in how they couple velocity indices.

Classification by velocity coupling

From a numerical standpoint, the collision operators fall into three categories:

  • Local in velocity
    Ionization acts independently at each velocity point.

  • Global in velocity (linear or bilinear)
    Charge exchange couples all velocity points through integrals or convolutions.

  • Global with nonlinear moment constraints
    BGK collisions require solving for equilibrium parameters that reproduce discrete moments.

This classification strongly influences computational cost and algorithmic design.

Electron-impact ionization

Model form

Ionization is modeled as a velocity-independent loss term,

\[ C_{\mathrm{iz}}[f_n](\mathbf{v}) = - \nu_{\mathrm{iz}} f_n(\mathbf{v}), \]

where the ionization frequency \(\nu_{\mathrm{iz}}\) depends on the local electron density and temperature through a Maxwellian-averaged rate coefficient.

Discrete form

In the DVM, ionization reduces to

\[ (C_{\mathrm{iz}})_j = - \nu_{\mathrm{iz}} f_{n,j}, \qquad j = 1,\dots,N_v. \]

No velocity integrals or coupling are required.

Numerical properties

  • Fully local in velocity and configuration space
  • Linear operator
  • Computational cost scales as \(\mathcal{O}(N_v)\)
  • Trivially parallel over velocity indices

Implementation in GUERNICA

Ionization operator

  • IonizationOperator.hxx / IonizationOperator.cxx

Ionization is applied pointwise to each velocity-resolved DG field using externally prescribed rate coefficients.

Charge exchange

Model form

Charge exchange between neutrals and ions is modeled using a symmetric gain–loss operator,

\[ \begin{aligned} C_{\mathrm{cx}}[f_n](\mathbf{v}) = {} & - f_n(\mathbf{v}) \int f_i(\mathbf{v}') \sigma_{\mathrm{cx}}(|\mathbf{v}-\mathbf{v}'|) |\mathbf{v}-\mathbf{v}'|\, d\mathbf{v}' \\ & + f_i(\mathbf{v}) \int f_n(\mathbf{v}') \sigma_{\mathrm{cx}}(|\mathbf{v}-\mathbf{v}'|) |\mathbf{v}-\mathbf{v}'|\, d\mathbf{v}' . \end{aligned} \]

Here \(f_i\) is a Maxwellian ion distribution with prescribed density, flow velocity, and temperature.

Defining the kernel

\[ K_{\mathrm{cx}}(|\mathbf{v}-\mathbf{v}'|) = \sigma_{\mathrm{cx}}(|\mathbf{v}-\mathbf{v}'|) |\mathbf{v}-\mathbf{v}'|, \]

the operator can be written in convolution form.

Discrete velocity formulation

In the DVM, velocity integrals are replaced by quadrature sums,

\[ (K_{\mathrm{cx}} \ast f)_j = \sum_{k=1}^{N_v} K_{\mathrm{cx}}(|\mathbf{v}_j-\mathbf{v}_k|) f_k w_k. \]

The discrete charge exchange operator becomes

\[ (C_{\mathrm{cx}})_j = - f_{n,j} (K_{\mathrm{cx}} \ast f_i)_j + f_{i,j} (K_{\mathrm{cx}} \ast f_n)_j. \]

This structure introduces global velocity coupling.

Numerical realizations

GUERNICA provides multiple numerical realizations of the charge exchange operator:

Direct quadrature

  • Evaluate the convolution explicitly
  • Computational cost \(\mathcal{O}(N_v^2)\)
  • Used primarily for verification and small velocity grids

Meier approximation

  • Approximate the convolution kernel using a reduced model
  • Lower cost than direct quadrature
  • Reduced accuracy in strongly non-Maxwellian regimes

FFT-based convolution

  • Exploit the convolution structure on uniform Cartesian velocity grids
  • Computational cost \(\mathcal{O}(N_v \log N_v)\)
  • Dominant approach for large 3D velocity grids

Implementation in GUERNICA

Charge exchange operators

  • ChargeExchangeOperator.hxx / .cxx — 1V test cases
  • ChargeExchangeFull.hxx / .cxx — 3V direct quadrature
  • ChargeExchangeMeier.hxx / .cxx — Meier approximation
  • ChargeExchangeFFT.hxx / .cxx — FFT-accelerated convolution

The FFT formulation relies on uniform velocity spacing and tensor-product structure, as described in the DVM section.

Neutral–neutral collisions (BGK model)

Neutral–neutral elastic collisions are modeled using a BGK approximation, which relaxes the neutral distribution toward a local equilibrium while conserving mass, momentum, and energy.

Model form

The BGK collision operator is written as

\[ C_{\mathrm{bgk}}[f_n](\mathbf{v}) = - \nu_{\mathrm{nn}} \left( f_n(\mathbf{v}) - f_{\mathrm{eq}}(\mathbf{v}) \right), \]

where \(\nu_{\mathrm{nn}}\) is an effective neutral–neutral collision frequency and \(f_{\mathrm{eq}}\) is a Maxwellian equilibrium distribution with the same density, momentum, and energy as \(f_n\).

The BGK operator is local in configuration space but global in velocity space, since the equilibrium depends on velocity moments of the full distribution.

Discrete equilibrium representation

In the discrete velocity method, the equilibrium distribution is represented in exponential form,

\[ f_{\mathrm{eq},j} = \exp\!\left( \alpha_0 + \boldsymbol{\alpha}_1 \cdot \mathbf{v}_j + \alpha_2 |\mathbf{v}_j|^2 \right), \qquad j = 1,\dots,N_v. \]

The coefficients \(\alpha_0\), \(\boldsymbol{\alpha}_1\), and \(\alpha_2\) are chosen such that the discrete velocity moments of \(f_{\mathrm{eq}}\) match those of \(f_n\).

Discrete moment constraints

The equilibrium coefficients are determined by enforcing the following constraints:

\[ \sum_j f_{\mathrm{eq},j} w_j = \sum_j f_{n,j} w_j, \]
\[ \sum_j \mathbf{v}_j f_{\mathrm{eq},j} w_j = \sum_j \mathbf{v}_j f_{n,j} w_j, \]
\[ \sum_j |\mathbf{v}_j|^2 f_{\mathrm{eq},j} w_j = \sum_j |\mathbf{v}_j|^2 f_{n,j} w_j. \]

These equations enforce discrete conservation of mass, momentum, and energy and form a nonlinear system for the equilibrium coefficients at each configuration-space degree of freedom.

Nonlinear solve for equilibrium coefficients

The nonlinear system for \(\{\alpha_0, \boldsymbol{\alpha}_1, \alpha_2\}\) is solved using a Newton iteration following the approach of Mieussens.

At each Newton step:

  1. The equilibrium distribution \(f_{\mathrm{eq},j}\) is evaluated at all velocity points.
  2. Discrete moments of \(f_{\mathrm{eq}}\) are computed using the same velocity quadrature as the solver.
  3. The residual between target and equilibrium moments is formed.
  4. The Jacobian of the moment system with respect to the coefficients is assembled and solved.

Because the system is low-dimensional (five unknowns in 3D velocity space), the Newton solve is inexpensive relative to transport or charge exchange.

Application of the BGK operator

Once the equilibrium distribution is constructed, the BGK operator is applied pointwise in velocity space:

\[ (C_{\mathrm{bgk}})_j = - \nu_{\mathrm{nn}} \left( f_{n,j} - f_{\mathrm{eq},j} \right). \]

The operator is linear in \(f_n\) once the equilibrium has been computed, but nonlinear overall due to the moment-matching step.

Numerical properties

From a numerical standpoint, the BGK operator has the following characteristics:

  • Global coupling in velocity space through moment constraints
  • Exact discrete conservation of mass, momentum, and energy
  • Computational cost \(\mathcal{O}(N_v)\) per configuration-space degree of freedom
  • Nonlinear but low-dimensional solve per spatial DOF

Implementation in GUERNICA

BGK operator

  • BGKOperator.hxx / BGKOperator.cxx

The BGK operator computes discrete velocity moments, solves the nonlinear equilibrium system via Newton iteration, and applies relaxation toward the discrete equilibrium at each configuration-space degree of freedom.

Summary

From a numerical perspective:

  • Ionization is local, linear, and inexpensive
  • Charge exchange dominates computational cost and motivates FFT acceleration
  • BGK collisions require nonlinear solves to enforce discrete conservation

The design of GUERNICA’s collision operators reflects a balance between physical fidelity, numerical consistency, and computational efficiency.