High-Order Asymptotic-Preserving IMEX schemes for an ES-BGK model for Gas Mixtures

TL;DR

The paper tackles multiscale kinetic modeling of gas mixtures by developing a high-order asymptotic-preserving IMEX scheme for the multi-species ES-BGK model. It combines a third-order IMEX Runge-Kutta time discretization with CWENO3 spatial reconstruction, a conservative velocity discretization, and a parallel implementation to handle arbitrary numbers of species. The scheme remains uniformly stable across Knudsen numbers, recovers the correct Euler (and Navier–Stokes in the appropriate limit) behavior in the fluid regime, and shows strong accuracy in kinetic regimes through a comprehensive suite of tests, including comparisons to BGK and full Boltzmann operators. This work provides a robust, efficient framework for simulations of gas mixtures across regimes, with potential extensions to more complex collisions and mesh types.

Abstract

In this work we construct a high-order Asymptotic-Preserving (AP) Implicit-Explicit (IMEX) scheme for the ES-BGK model for gas mixtures introduced in [Brull, Commun. Math. Sci., 2015]. The time discretization is based on the IMEX strategy proposed in [Filbet, Jin, J. Sci. Comput., 2011] for the single-species BGK model and is here extended to the multi-species ES-BGK setting. The resulting method is fully explicit, uniformly stable with respect to the Knudsen number and, in the fluid regime, it reduces to a consistent and high-order accurate solver for the limiting macroscopic equations of the mixture. The IMEX structure removes the stiffness associated with the relaxation term so that the time step is constrained only by a hyperbolic CFL condition. The full solver couples a high-order space and velocity discretization that includes third-order time integration, a CWENO3 finite-volume reconstruction in space, exact conservation of macroscopic moments in the discrete velocity space, and a multithreaded implementation. The proposed approach can handle an arbitrary number of species. Its accuracy and robustness are demonstrated on a set of multidimensional kinetic tests for gas mixtures, where the AP property and the correct asymptotics are numerically verified across different regimes.

Figures (8)

• Figure 1: Moments evolution for the homogeneous relaxation \ref{['Test00']}. In the first row it is reported the evolution of the temperature and the velocity of each species for a mass ratio equal to 1. The same quantities, but for a mass ratio equal to 100, are reported in the second row. The solid line in each figure represents the asymptotic value associated to the corresponding equilibrium Maxwellian given by equation \ref{['eq7']}, while the crosses and the open circles represents, respectively, the first and the second species.
• Figure 2: Convergence rate in time for different values of $\varepsilon$ for the time convergence \ref{['Test01']}.
• Figure 3: Convergence rate in space and time for different values of $\varepsilon$ for the space and time convergence \ref{['Test01']}.
• Figure 4: Total density, temperature and velocity for Sod Shock Tube problem (\ref{['Test1']}), for different values of Knudsen number at the final time $t=0.15$ and with unitary mass ratio.
• Figure 5: Total Density, Pressure and Velocity in the Sod Shock Tube problem (\ref{['Test1']}), for different values of Mass Ratio ($MR=m_2/m_1$), with a Knudsen number equal to $\varepsilon=10^{-6}$, at the final time $t=0.15$.