Asymptotic preserving methods for the low mach limit in discrete velocity models approximating kinetic equations

TL;DR

The paper addresses the challenge of simulating low-Mach, multiscale flows that transition from kinetic to incompressible hydrodynamics. It introduces an asymptotic-preserving framework based on a lattice Boltzmann discrete velocity model with diffusive scaling, paired with IMEX Runge-Kutta time stepping and high-order spatial discretization (including WENO), to maintain uniform stability across Knudsen and Mach regimes. In the vanishing-scale limit ($oldsymbol{ o}0$), the scheme naturally reduces to a high-order projection method for the incompressible Navier–Stokes equations, while remaining accurate for finite relaxation parameters. Numerical tests in 2D confirm accuracy, stability, and the method’s ability to capture complex flow features across both kinetic and hydrodynamic regimes, validating the AP approach and its potential for extension to more general Boltzmann-type models.

Abstract

We consider a Lattice Boltzmann type discrete velocity model in the low Mach number scaling and develop a corresponding numerical scheme that remains uniformly valid across all regimes of the mean free path, from the kinetic to the hydrodynamic scale. The proposed framework ensures high order temporal accuracy through the use of Implicit Explicit Runge Kutta methods, which provide stability and efficiency in stiff regimes, while spatial resolution is enhanced by combining finite difference WENO reconstructions with high order central difference approximations. In the appropriate asymptotic limit, the scheme reduces to a high order finite difference formulation of the incompressible Navier Stokes equations, thereby guaranteeing physical consistency of the numerical approximation with the limit model. To corroborate the theoretical findings, a set of numerical experiments is performed on two dimensional benchmark problems, which confirm the accuracy, stability, and versatility of the method across different flow regimes.

Figures (7)

• Figure 1: Convergence rate for the second order IMEX-RK method in time and third order WENO reconstruction in space for different values of $\varepsilon$. (left panel: $\tau = 0$, right panel: $\tau = 0.05$)
• Figure 2: Thick shear layer test case in 2D with $\tau = 0$ (Incompressible Euler case) and $\varepsilon = 10^{-6}$. The vorticity (left panel) and the time history of the $L^\infty$ norm of the divergence (right panel) are shown at $t = 6$.
• Figure 3: Thick shear layer test case in 2D with $\tau = 0.05$ (Incompressible Navier-Stokes case) and $\varepsilon = 10^{-6}$. The vorticity (left panel) and the time history of the $L^\infty$ norm of the divergence (right panel) are shown at $t = 9.5$.
• Figure 4: Thick shear layer test case in 2D with $\varepsilon = 0.25$. The incompressible Euler case with $\tau = 0$ (left panel) is shown at $t = 6$ and the INS case with $\tau = 0.05$ (right panel) is shown at $t = 9.5$.
• Figure 5: Thin shear layer test case in 2D with $\tau = 0$ (Incompressible Euler case) and $\varepsilon = 10^{-6}$. The vorticity (left panel) and the time history of the $L^\infty$ norm of the divergence (right panel) are shown at $t = 6$.

Theorems & Definitions (7)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• proof
• Theorem 3.5
• proof