A Spectral Approach to Multiscale Modeling with Coupled Hydrodynamic and Kinetic Effects

We present a multiscale method which leverages spectral techniques to model flows in regimes where atomistic effects are important, but at scales where kinetic calculations are not possible. The algorithm solves the multispecies Boltzmann equation with a unified discretization across near-continuum and kinetic-dominated regions. For simplicity, collisions are modeled with the Bhatnagar-Gross-Krook operator, although generalizations are straightforward. The velocity space is discretized by projecting onto asymmetrically weighted Hermite basis functions and the resulting system of equations for the spectral coefficients is solved using high order finite differences in a highly scalable implementation. The number of terms in the spectral expansion is adapted in space and time as needed to capture the relevant physics. Near-continuum regions require only the first few terms, while regions with strong kinetic effects require more terms in the expansion. We present verification of the algorithm, demonstrating computational efficiency, mathematical correctness, and physics simulation capabilities with canonical examples.