Linear Kinetic BGK Models

• Linear kinetic BGK models are a class of kinetic equations that approximate the Boltzmann equation using linearized collision operators and conserve mass, momentum, and energy.
• They facilitate detailed analysis of hypocoercivity and exponential relaxation through explicit spectral methods and high-order numerical schemes.
• The models support multiscale domain decomposition and uncertainty quantification in kinetic-fluid coupled simulations, enhancing the robustness of gas mixture studies.

The linear kinetic BGK (Bhatnagar-Gross-Krook) models are a central class of kinetic equations used to approximate the Boltzmann equation for the evolution of one-particle distribution functions in phase space. The linear BGK models capture essential features of collision-driven relaxation to equilibrium while enabling analytical and numerical tractability via linearized collision operators. These models are fundamental for studying hypocoercivity, error amplification due to modelling uncertainties, high-order discretizations, and the coupling of kinetic equations with macroscopic fluid models.

1. Structure of Linear BGK Kinetic Equations

The linear BGK equation typically takes the form

$$\partial_t f + v \cdot \nabla_x f = \frac{1}{\mathrm{Kn}} (M[f] - f),$$

where $f = f(t, x, v)$ is the phase-space distribution, $v$ is the velocity, $\mathrm{Kn}$ is the Knudsen number, and $M[f]$ is the local Maxwellian determined by conservation of mass, momentum, and energy via the relevant moments of $f$. Linearization proceeds by expanding around a background Maxwellian $M_0(v; u_0, T_0)$ and setting $f = M_0 (1+h)$, assuming $|h| \ll 1$. Omitting higher-order terms yields

$$\partial_t h + v \cdot \nabla_x h = \mathcal{L}[h],$$

where the linearized BGK operator is

$\mathcal{L}[h] = \nu (P_0[h] - h),$

with $P_0$ being the orthogonal projector onto the collision invariants $\operatorname{Span}\{1, v, |v|^2\}$ in the weighted space $L^2(M_0^{-1} dv)$ (Klingenberg et al., 2018).

In gas mixtures, the linearization yields coupled systems for each component. For a two-species mixture, the equations are

$$\partial_t h_k + v \cdot \nabla_x h_k = L_{k1} h_1 + L_{k2} h_2, \quad k=1,2,$$

with interspecies and intraspecies collision terms governed by linearized operators and projectors derived from the mixture’s Maxwellians (Liu et al., 2018).

2. Collision Operators, Conservation Laws, and Spectral Properties

The BGK collision operator in linearized form satisfies

$\mathcal{L}[h](v) = \nu (P_0[h] - h),$

where $P_0$ is the orthogonal projection in $L^2(M_0^{-1} dv)$. The operator is negative semi-definite and dissipative outside the kernel spanned by the hydrodynamic (collision invariant) modes; the collision term preserves moments corresponding to mass, momentum, and energy, enforcing these conservation constraints (Grosse, 20 Feb 2025).

In mixtures, the collision matrix has block-structure with self and cross terms. Cross-Maxwellians $M_{12}$, $M_{21}$ yield coupling matrices in the kinetic system, introducing additional complexity for energy and momentum conservation at the mixture level. The invariant (null) space is spanned by Maxwellian-weighted polynomials (the collision invariants), and the spectral gap ensures rapid relaxation of kinetic components orthogonal to the invariants (Liu et al., 2018).

3. Hypocoercivity and Exponential Relaxation

Linear kinetic BGK models display hypocoercivity, reflecting the combination of dissipative collision dynamics and conservative transport evolution. This structure allows for exponential decay of perturbations to equilibrium.

For a one-species model, Fourier/Hermite mode expansions and the construction of modified (Lyapunov) entropy functionals yield explicit exponential decay rates. In mixtures, the entropy combines the contributions from all species, and the coupling terms introduce off-diagonal components controlled via Cauchy–Schwarz/Young inequalities. Under normalization and conservation constraints, solutions satisfy

$$e(h_1(t), h_2(t)) \leq e^{-\Lambda t} e(h_1(0), h_2(0)),$$

with explicit $\Lambda$ depending on mixture parameters and collision frequencies (Liu et al., 2018).

In fully spectral numerical schemes, the hypocoercive entropy can be discretized to show that the semi-discrete solution decays exponentially at a rate independent of the truncation in velocity or space. All continuous invariants (mass, momentum, energy) are preserved, and spectral accuracy is observed numerically (Grosse, 20 Feb 2025).

4. Numerical Methods for Linear BGK Equations

Several high-order numerical strategies have been developed for linear BGK kinetics.

• Spectral Methods: These use Hermite polynomials in velocity and orthonormal polynomials in space (w.r.t. an equilibrium measure) for basis expansion. Projecting the kinetic equation onto this basis and truncating yields semi-discrete ODE systems with exact conservation of mass, momentum, and energy. The method preserves hypocoercivity and shows exponential decay at the discrete level for polynomial confining potentials (Grosse, 20 Feb 2025).
• Domain Decomposition and Coupling: In multi-scale modeling, a domain-decomposition approach is used, where Euler equations (fluid regime) and BGK kinetic solvers (kinetic regime) are coupled via the solution of linearized half-space problems (the Knudsen layer). Spectral solvers for the half-space provide the correct boundary-layer transfer between regimes. The entire algorithm achieves $L^2$ error scaling as $O(\epsilon)$ between the BGK solution and the coupled Euler–layer method (Chen et al., 2018).
• Fully Explicit High-Order Kinetic Schemes with Matrix Collisions: BGK-based kinetic schemes can target arbitrarily prescribed convection–diffusion systems (including for nonlinear, multi-component systems) by replacing the BGK relaxation by a matrix collision operator. Chapman–Enskog expansions guarantee asymptotic accuracy up to $O(\varepsilon^2)$ and allow arbitrary order in time and space via deferred-correction Lobatto schemes. The approach maintains a hyperbolic CFL condition, avoiding parabolic time-step restrictions (Wissocq et al., 2023).

5. Quantification of Modelling Uncertainties

The sensitivity of the linearized BGK solution to the choice of equilibrium Maxwellian is quantified rigorously. Linearizing about an inaccurate Maxwellian, with uncertainties $\delta u$ (velocity) and $\delta T$ (temperature), leads to solution errors that are Lipschitz in these parameters. Explicitly,

$$\| h_\delta(t) - h_0(t) \|_{L^2(M_0^{-1})} \leq ( \| h_\delta(0) - h_0(0) \| + C(|\delta u| + |\delta T|) ) e^{-\lambda_0 t} + C (|\delta u| + |\delta T|),$$

with constants dependent on the background parameters. For time-stepping schemes, local updates to the Maxwellian must maintain closeness to avoid physical inaccuracies. For randomized or stochastic approaches (e.g., generalized polynomial chaos), the error bounds imply only algebraic convergence in the presence of such model uncertainties (Klingenberg et al., 2018).

6. Mathematical and Practical Implications

Linear kinetic BGK models are foundational for both theoretical and computational kinetic theory. They enable explicit analysis of relaxation rates, stability under model perturbation, and the development of high-order, conservative, and hypocoercive-preserving numerical schemes. Key mathematical outcomes include:

• Rigorous exponential convergence to equilibrium under explicit, computable rates for mixtures and single-component gases (Liu et al., 2018).
• Demonstration of the robustness of the linearized BGK operator to small perturbations in equilibrium parameters (Klingenberg et al., 2018).
• High-fidelity numerical schemes capable of solving the linear BGK equation in complex potentials, coupled regimes, and multi-component settings, with explicit preservation of conservation laws and hypocoercivity at the discrete level (Grosse, 20 Feb 2025, Chen et al., 2018, Wissocq et al., 2023).

The principles underlying linear kinetic BGK models also underpin multiscale domain decomposition algorithms for micro–macro coupling and inform the error analysis and design of uncertainty quantification frameworks. These models thus serve as the analytical and computational backbone of contemporary kinetic theory research and its applications in rarefied flows, gas mixtures, and kinetic–fluid couplings.