Locally adaptive spectral method for multiphysics kinetic Boltzmann equation

Spectral methods based on asymmetrically-weighted Hermite polynomials can efficiently quantize velocity space for problems encompassing near-continuum and kinetic regions. We present spatially and temporally adaptive physics-based algorithm for Hermite polynomials applied to Vlasov-Maxwell and Boltzmann-BGK systems. The spectral adaptivity enables accurate solutions of local flow velocity and pressure fields with lower number of spectral coefficients. It acts as a regularization technique to the spectral approach for problems with high variations in pressure and/or flow velocities, significantly improving its stability properties. For the verification of the algorithm, we have performed convergence studies with the method of manufactured solutions, testing both temporal and spatial adaptivity. Finally, we demonstrate the capabilities of the method on Whistler instability, Orszag-Tang vortex dynamics, and Sedov blast wave, comparing against the standard non-adaptive approach, and discuss the conservation properties.