A Modified BGK Collision Operator for Exact Conservation in Numerical Solutions of Boltzmann-BGK

TL;DR

This paper introduces a novel numerical method for solving the 1D1V Boltzmann-BGK equation that guarantees exact conservation of mass, momentum, and energy, improving accuracy and robustness across all flow regimes.

Contribution

The authors develop a new collision operator treatment ensuring exact conservation in numerical solutions of the Boltzmann-BGK equation, combining operator splitting, high-order schemes, and a quadratic Hermite polynomial correction.

Findings

Exact conservation of mass, momentum, and energy demonstrated

Method verified on piecewise constant initial data with periodic boundaries

Implementation available as open-source Java code

Abstract

Ideal gases can be modeled by the Boltzmann equation from statistical physics. Instead of trying to track the position and velocity of a large number of gas molecules, it is possible to describe the particles with a particle distribution function. The Boltzmann equation provides the rule for evolving the distribution function over time, allowing one to simulate the gas dynamics. In this work, we develop a novel numerical method for solving the 1D1V Boltzmann-BGK equation. Several important ingredients are combined to create an accurate, efficient, robust numerical method valid in all flow regimes. First, we make use of operator splitting to create separate transport and collision sub-steps, each of which is easier to discretize than the whole system; second, we introduce a third-order accurate Lax-Wendroff-type scheme for the transport sub-step; third, we make use of the second-order L-stable TR-BDF method for the collision sub-step. The final component we introduce is a novel treatment of the collision sub-step to guarantee that the total mass, momentum, and energy are conserved even in the presence of a truncated velocity range and quadrature errors in computing moments. The key idea is to multiply the Maxwell-Boltzmann distribution in the collision sub-step by a quadratic Hermite polynomial, where the coefficients in the polynomial are chosen to ensure exact conservation. The resulting scheme is verified on piecewise constant initial data with periodic boundary conditions; exact conservation up to machine precision is demonstrated for mass, momentum, and energy. The resulting method is implemented in a freely available Java code with Python plotting routines.