Adaptivity of spectral method for multiphysics kinetic Boltzmann solver

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 also 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 verification, we have performed convergence studies with the method of manufactured solutions. We further demonstrate the capabilities of the method on Whistler instability, Orszang-Tang vortex dynamics, and Sedov blast wave, comparing against the standard non-adaptive approach, and discuss the conservation properties.

Moreover, we exploit the properties of advection operator in spectral space to modify existing shock-capturing schemes (e.g. WENO) to be fully compatible with spectral adaptivity. A benchmark problem example is a multiscale Sedov blast wave, where collisional time and shock thickness are separated by many orders of magnitude from advective time and problem scale, respectively. To this end, we apply previously developed implicit-explicit time integration scheme to address temporal stiffness and couple it with the new shock-capturing scheme with spectral adaptivity for spatial dynamics.