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Non-dimensionalization

Non-dimensionalization plays a critical role in the numerical stability and robustness of kinetic simulations. In GUERNICA, careful scaling is used to reduce the dynamic range of physical quantities, improve conditioning of discrete operators, and enable stable time integration across disparate physical regimes.

This page describes the reference scales used in GUERNICA and how the governing equations are non-dimensionalized prior to discretization.

Motivation

Neutral kinetic simulations in fusion-relevant regimes involve quantities that span many orders of magnitude:

  • densities varying from \(10^{14}\)–\(10^{20}\,\mathrm{m}^{-3}\),
  • velocities on the order of \(10^4\)–\(10^5\,\mathrm{m/s}\),
  • temperatures from fractions of an eV to tens of eV,
  • collision frequencies ranging from nearly collisionless to strongly collisional.

Without non-dimensionalization, these scales can lead to:

  • poorly conditioned mass and stiffness operators,
  • loss of numerical precision,
  • overly restrictive time-step constraints.

Non-dimensionalization mitigates these issues by expressing all quantities in terms of characteristic reference scales.

Reference scales

GUERNICA introduces characteristic reference quantities:

  • Reference length: \(L_0\)
  • Reference velocity: \(V_0\)
  • Reference time: \(T_0 = L_0 / V_0\)
  • Reference density: \(n_0\)
  • Reference distribution function:
    \( f_0 = \frac{n_0}{V_0^3} \)

These scales are chosen to be representative of the physical problem under consideration and are typically derived from background plasma parameters.

Dimensionless variables

Physical variables are expressed in dimensionless form as:

\[ \tilde{\mathbf{x}} = \frac{\mathbf{x}}{L_0}, \qquad \tilde{t} = \frac{t}{T_0}, \qquad \tilde{\mathbf{v}} = \frac{\mathbf{v}}{V_0}, \]
\[ \tilde{f}_n = \frac{f_n}{f_0}. \]

All equations are written in terms of dimensionless quantities, with tildes omitted for clarity in the remainder of the documentation.

Non-dimensionalized kinetic equation

Applying these scalings to the neutral Boltzmann equation yields

\[ \frac{\partial f_n}{\partial t} + \mathbf{v} \cdot \nabla f_n = \frac{L_0}{V_0} \left( C_{\mathrm{iz}}[f_n] + C_{\mathrm{cx}}[f_n] + C_{\mathrm{bgk}}[f_n] \right). \]

The transport term becomes order unity, while collision operators are weighted by dimensionless parameters reflecting their relative strength.

Dimensionless collision frequencies

Each collision operator introduces a dimensionless parameter of the form

\[ \hat{\nu} = \frac{L_0}{V_0} \nu, \]

where \(\nu\) is the physical collision frequency.

These parameters determine whether the simulation is transport-dominated or collision-dominated.

Ionization

\[ \hat{\nu}_{\mathrm{iz}} = \frac{L_0}{V_0} \nu_{\mathrm{iz}}(n_e, T_e). \]

Charge exchange

Charge exchange introduces a dimensionless kernel

\[ \hat{K}_{\mathrm{cx}}(|\mathbf{v}-\mathbf{v}'|) = \frac{L_0}{V_0} \sigma_{\mathrm{cx}}(|\mathbf{v}-\mathbf{v}'|) |\mathbf{v}-\mathbf{v}'|. \]

This scaling is applied consistently in all numerical realizations of the operator.

BGK collisions

The BGK collision frequency becomes

\[ \hat{\nu}_{\mathrm{nn}} = \frac{L_0}{V_0} \nu_{\mathrm{nn}}. \]

This parameter directly controls the stiffness of the BGK relaxation term in time integration.

Impact on DG discretization

Non-dimensionalization affects the DG discretization in several important ways:

  • Mass matrix entries scale uniformly with the reference volume, improving conditioning.
  • Transport operators operate on order-unity velocities.
  • Collision operators are naturally compared to transport through dimensionless parameters.

In particular, non-dimensionalization improves the conditioning of the element-local mass matrix, which is inverted iteratively using CG.

Velocity-space truncation and scaling

Velocity space is truncated to a finite domain,

\[ \mathbf{v} \in [-v_{\max}, v_{\max}]^d, \]

where \(v_{\max}\) is specified in dimensionless units. Choosing \(v_{\max}\) of order a few thermal speeds ensures:

  • adequate resolution of the distribution tail,
  • stable time stepping,
  • controlled CFL constraints.

Interpretation of results

Simulation outputs are stored in dimensionless form. Physical units are recovered through inverse scaling:

\[ \mathbf{x} = L_0 \tilde{\mathbf{x}}, \qquad \mathbf{v} = V_0 \tilde{\mathbf{v}}, \qquad t = T_0 \tilde{t}, \qquad f_n = f_0 \tilde{f}_n. \]

Derived quantities such as density, flow velocity, and temperature are rescaled consistently using these relations.

Summary

Non-dimensionalization in GUERNICA:

  • reduces the dynamic range of physical quantities,
  • improves numerical conditioning,
  • clarifies the relative importance of transport and collisions,
  • enables stable time integration across regimes.

This scaling framework is applied consistently throughout configuration-space discretization, velocity-space integration, and collision operator evaluation.