Asymptotic preserving schemes for the kinetic Boltzmann-ES-BGK equation

Abstract

A rarefied gas is defined as a gas in which the mean free path of the molecules is large compared to the size of the bodies present, leading to molecules moving freely without frequent collisions, affecting processes like heat transfer differently than in dense gases. There are many applications of kinetic theory in which rarefaction effects play a significant role. A critical parameter called the Knudsen number (ε = λ/L) characterizes the degree of rarefaction, where λ is the average distance traveled by the molecules between collisions, or the mean free path, and L is the characteristic length scale. An important field of application of kinetic theory is gas flow problems involving large mean free paths in high-altitude flows. For example, the motion of objects in the rarefied layers of the outer atmosphere, such as spacecraft re-entry problems in aerospace engineering. Here, the Knudsen number is large, because the mean free path is of orders of magnitude larger than the characteristic length of the space vehicle. A huge new field of applications of Boltzmann equations has developed in the modeling of fluid flows in nanostructures. In this case, the Knudsen number is large because the scale of interest L is so small that the ratio λ/L is of order one and microscopic effects cannot be ignored. Accurate descriptions of fluid flows in the transitional continuum regime is of increasing technological relevance because of the growing trend toward miniaturization, for instance, in nanoscale applications, micro-channel flows or flow in porous media.

The Boltzmann equation, one of the most well-known kinetic models for rarefied gases, describes the motion of a fluid for the simulation of gas dynamics over a wide range of Knudsen numbers. An important class of methods are so-called asymptotic-preserving (AP) schemes, which allow the numerical method to be stable at fixed mesh parameters for any value of the Knudsen number, including in the fluid (very small Knudsen numbers), slip flow (small Knudsen numbers), transition (moderate Knudsen numbers), and free molecular flow (large Knudsen numbers) regimes. To avoid the complexity of the Boltzmann collision operator, the Bhatnagar-Gross-Krook (BGK) collision operator and the closely related Ellipsoidal Statistical BGK (ES-BGK) collision operator is used, which, unlike BGK, gives the correct transport coefficients. In this work, we develop asymptotic preserving numerical schemes for solving the Boltzmann ES-BGK equation and implement them in parallel with MPI for efficiency. These schemes are uniformly stable as we transition from the kinetic regime to the fluid regime and are applied to various test cases in 1D and 2D.

The first scheme we consider is the direct kinetic scheme, which is implicit in the source term and explicit in the transport term. This method is second order accurate in both space and time. Here we try to solve the full distribution function over space and velocity. We can find the macroscopic variables by finding moments of this distribution function. We apply this scheme to the 1D shock tube problem and the 2D lid driven cavity problem and then analyze the results. We then parallelize our scheme using MPI and study the efficiency of the parallelized scheme by doing weak scaling and strong scaling with varying number of processors and study the results. This scheme is also applied to another two-dimensional test case such as the thermally induced edge flow in a hot beam inside a cold chamber.

The second scheme we consider is the micro-macro decomposition scheme, which couples a microscopic kinetic equation with a macroscopic fluid equation. In this method, the Boltzmann equation is equivalently written as a system coupling a hydrodynamic part with a kinetic part of the distribution function. A projection operator is used to separate the macroscopic and microscopic quantities. The updated microscopic quantity is coupled to the macroscopic update equation via the heat flux variable, which is obtained by taking the moment of the microscopic distribution function. The updated macroscopic variables are then used in the calculation of the projection operator which affect the microscopic update. Again, this scheme is parallelized and tested on 1D and 2D test problems.

Finally, we consider an adaptive version of the micro-macro scheme in order to make it more efficient. We present an adaptive solver that works mainly in the fluid regime on a coarse velocity mesh and can adapt to boundary layers and or shocks by adaptively increasing velocity resolution in the microscopic variable. This method is shown to be computationally more efficient than the direct kinetic scheme and the non-adaptive micro-micro schemes on stationary 1D test cases with boundary layers and shocks.