Micro-macro kinetic flux-vector splitting schemes for the multidimensional Boltzmann-ES-BGK equation

The kinetic Boltzmann equation models gas dynamics over a wide range of spatial and temporal scales. Simplified versions of the full Boltzmann collision operator, such as the classical Bhatnagar-Gross-Krook and the closely related Ellipsoidal-Statistical-BGK operators, can dramatically decrease the computational costs of numerical solving kinetic equations. Classical BGK yields incorrect transport coefficients (relative to the full Boltzmann collision operator) at low Knudsen numbers, whereas ES-BGK captures them correctly. In this work, we develop a finite volume method using a micro-macro decomposition of the distribution function, which requires a smaller velocity mesh relative to direct kinetic methods for low and intermediate Knudsen numbers. The macro portion of the model is a fluid model with a moment closure provided from the heat flux tensor calculated from the micro portion. The micro portion is obtained by applying to the original kinetic equation a projector into the orthogonal complement of the null space of the collision operator - this projector depends on the macro portion. In particular, we extend the technique of Bennoune, Lemou, and Mieussens [Uniformly stable schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys. (2008)] to two-space dimensions, the ES-BGK collision operator, and problems with reflecting wall boundary conditions. As it appears in both the micro and macro equations, the collision operator is handled via L-stable implicit time discretizations. At the same time, the remaining transport terms are computed via kinetic flux vector splitting (for macro) and upwind differencing (for micro). The resulting scheme is applied to various test cases in 1D and 2D. The 2D version of the code is parallelized via MPI, and we present weak and strong scaling studies with varying numbers of processors.