High-Order Asymptotic-Preserving Schemes for Kinetic Equations from Rarefied to Incompressible Regimes

Technical Summary: High-Order Asymptotic-Preserving Schemes for Kinetic Equations

Problem Statement
The paper addresses the numerical challenge of solving kinetic equations, specifically the BGK model, across all regimes of the mean free path, ranging from the rarefied kinetic scale to the incompressible hydrodynamic limit. A primary difficulty arises in the low Mach number limit, where fast acoustic waves decouple from slow convective modes, creating a "stiffness" issue. Standard numerical schemes that treat the small mean free path implicitly often require prohibitively small time steps in the low Mach limit. Furthermore, traditional stochastic methods like Direct Simulation Monte Carlo (DSMC) suffer from large statistical fluctuations when macroscopic velocities are negligible compared to thermal velocities. The objective is to develop a deterministic, high-order numerical framework that remains uniformly valid and accurate across these regimes without requiring mesh refinement as the scaling parameter (Knudsen number, $\epsilon$) tends to zero.

Methodology
The proposed framework is built upon a micro-macro decomposition, which reformulates the kinetic equation into a coupled system comprising:

1. A macroscopic part representing fluid-dynamic evolution (density, velocity, temperature).
2. A microscopic part describing non-equilibrium deviations.

The numerical strategy integrates the following components:

• Time Discretization: The method employs Implicit-Explicit (IMEX) Runge-Kutta (RK) schemes. Stiff collision operators are treated implicitly to ensure stability in stiff regimes, while transport terms are handled explicitly to maintain computational efficiency. The authors utilize Generalized Stiffly Accurate (GSA) IMEX-RK methods of the second and third orders.
• Spatial Discretization: To achieve high-order accuracy and non-oscillatory behavior, the scheme combines Weighted Essentially Non-Oscillatory (WENO) reconstructions for convective terms with high-order central difference approximations for diffusive terms.
• Velocity Space Discretization: The continuous velocity domain is discretized using the Discrete Velocity Method (DVM) with Gauss-Hermite quadrature. This choice ensures the exact recovery of equilibrium moments and minimizes the "curse of dimensionality."
• Asymptotic-Preserving (AP) Property: A critical feature is the design of the scheme such that, as $\epsilon \to 0$, it consistently degenerates into a high-order finite-difference formulation of the incompressible Navier-Stokes (INS) equations. The authors demonstrate that the discrete equations naturally reduce to a projection method for the INS equations in the hydrodynamic limit.
• Numerical Robustness: To handle severe cancellation errors in the computation of the second-order perturbation term $Q$ as $\epsilon \to 0$, the authors employ a Taylor expansion of the ratio of Maxwellians for small $\epsilon$, ensuring robustness in the hydrodynamic limit.

Key Contributions

1. Novel High-Order Framework: The paper introduces a family of numerical schemes that achieve uniform accuracy in the low Mach number limit without refining spatial or temporal discretization parameters.
2. Theoretical AP Proof: The authors provide a theoretical proof that the proposed IMEX-RK schemes, when applied to the micro-macro system, converge to a consistent high-order discretization of the incompressible Navier-Stokes equations (with heat conduction) as $\epsilon \to 0$.
3. Handling of Stiffness: The method successfully addresses the dual stiffness problems arising from small mean free paths and low Mach numbers, allowing for time steps restricted only by the macroscopic flow velocity and diffusion, rather than the microscopic relaxation time.
4. Algorithmic Implementation: The paper details the specific algorithmic steps (Algorithm 4.1) for solving the coupled system, including the treatment of the pressure Poisson equation and the specific handling of the collision term perturbation.

Results
Numerical experiments were conducted on one- and two-dimensional benchmark problems to validate the method:

• Convergence: In 1D smooth periodic tests, the method demonstrated the expected third-order convergence for both velocity and temperature, even for small $\epsilon$. While absolute errors for pressure were larger due to $O(1/\epsilon)$ scaling, relative errors remained low (approx. 0.3%).
• Comparison with BGK: In 1D tests involving sharp density transitions, the proposed AP scheme showed good agreement with a standard direct BGK solver across various $\epsilon$ and relaxation time ($\tau$) values.
• Computational Efficiency: The AP method significantly outperformed the standard BGK solver in the low $\epsilon$ regime. While the BGK solver's CPU time increased linearly with $1/\epsilon$ due to time-step restrictions, the AP method maintained time steps independent of $\epsilon$, resulting in substantial computational savings (e.g., 10x speedup when $\epsilon$ decreased from 1 to 0.1).
• 2D Shear Layer: In a 2D "thick shear layer" test, the high-order versions of the algorithm (using WENO3/WENO5) successfully captured peak amplitudes and steep gradients that were smoothed out by second-order approaches. The solver performed robustly in the rarefied, transitional, and macroscopic limits ($\epsilon = 10^{-6}$), matching incompressible Navier-Stokes and Euler limits as appropriate.

Significance
The paper claims that this work provides a robust, high-order, and asymptotic-preserving tool for simulating multiscale fluid phenomena. By bridging the gap between kinetic and hydrodynamic descriptions, the method allows for the simulation of flows where the mean free path varies significantly, avoiding the prohibitive costs of direct kinetic simulations in the continuum limit. The authors emphasize that the framework successfully recovers the correct fluid behavior (incompressible Navier-Stokes) in the appropriate limit while maintaining high-order accuracy, offering a versatile alternative to traditional stochastic or low-order deterministic methods. Future work, as noted by the authors, aims to extend this framework to non-periodic boundary conditions, the full Boltzmann collision operator, and irregular grids.