  It is challenging to solve the Boltzmann equation accurately due to the
extremely high dimensionality and nonlinearity. This paper addresses the idea
and implementation of the first flux reconstruction method for high-order
Boltzmann solutions. Based on the Lagrange interpolation and reconstruction,
the kinetic upwind flux functions are solved simultaneously within physical and
particle velocity space. The fast spectral method is incorporated to solve the
full Boltzmann collision integral with a general collision kernel. The explicit
singly diagonally implicit Runge-Kutta (ESDIRK) method is employed as time
integrator and the stiffness of the collision term is smoothly overcome.
Besides, we ensure the shock capturing property by introducing a self-adaptive
artificial dissipation, which is derived naturally from the effective cell
Knudsen number at the kinetic scale. As a result, the current flux
reconstruction kinetic scheme can be universally applied in all flow regimes.
Numerical experiments including wave propagation, normal shock structure,
one-dimensional Riemann problem, Couette flow and lid-driven cavity will be
presented to validate the scheme. The order of convergence of the current
scheme is clearly identified. The capability for simulating cross-scale and
non-equilibrium flow dynamics is demonstrated.

Computational Physics;Fluid Dynamics