Year-4 Progress Report on Numerical Methods for BGK-Type Kinetic Equations

This Year-Four progress report presents ongoing research, supported by subcontracts from Oak Ridge National Laboratory (ORNL), on the design and analysis of numerical methods for multi-species kinetic equations. Rather than summarizing only recent developments, this cumulative report builds upon prior work and serves as a resource for training new graduate students and postdoctoral researchers. Three new sections have been added. First, a comprehensive chapter on the multi-species Boltzmann equation addresses gaps in the literature and offers a modern, pedagogical introduction to this fundamental kinetic framework. Second, we introduce a novel iterative solver for moment equations derived from the space-homogeneous multi-species BGK model, laying the groundwork for more advanced numerical solvers. Third, we provide an overview of the Ellipsoidal Statistical BGK (ES-BGK) model, which improves upon the basic BGK model by better capturing fluid dynamics in the Navier–Stokes and Euler limits. The report also reviews recent progress on the numerical solution of BGK-type kinetic equations, highlighting the challenges posed by high-dimensional phase space, nonlocal particle interactions, and numerical stiffness. To address these, we describe strategies including high-order finite volume methods, implicit-explicit (IMEX) time integrators, adaptive multi-level methods, discrete velocity schemes, and moment-based approaches. Preliminary implementations are developed in MATLAB for prototyping, with plans to migrate to modern production software. The report concludes with a summary of achieved milestones and outlines directions for future work, particularly in developing robust solvers for the ES-BGK model.