A flux reconstruction kinetic scheme for the Boltzmann equation

TL;DR

This paper introduces a high-order flux reconstruction kinetic scheme for solving the Boltzmann equation, capable of handling all flow regimes and complex non-equilibrium dynamics with improved accuracy and efficiency.

Contribution

It presents the first flux reconstruction method for the Boltzmann equation, integrating spectral collision solutions and adaptive dissipation for broad applicability.

Findings

Achieves high-order accuracy in various flow regimes.

Demonstrates effective shock capturing and non-equilibrium flow simulation.

Validates the scheme through multiple numerical experiments.

Abstract

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.