Numerical Solver for the Boltzmann Equation With Self-Adaptive Collision Operators

TL;DR

This paper introduces a self-adaptive numerical solver for the Boltzmann equation that dynamically balances detailed binary collisions and approximate BGK collisions using an error indicator for improved efficiency.

Contribution

The paper presents a novel adaptive method that selects collision models locally and dynamically based on an error indicator, enhancing computational efficiency in solving the Boltzmann equation.

Findings

The error indicator effectively guides the adaptive collision model selection.

The method reduces computational cost while maintaining accuracy.

Numerical experiments validate the effectiveness of the adaptive approach.

Abstract

We use the Burnett spectral method to solve the Boltzmann equation whose collision term is modeled by separate treatments for the low-frequency part and high-frequency part of the solution. For the low-frequency part representing the sketch of the distribution function, the binary collision is applied, while for the high-frequency part representing the finer details, the BGK approximation is applied. The parameter controlling the ratio of the high-frequency part and the low-frequency part is selected adaptively on every grid cell at every time step. This self-adaptation is based on an error indicator describing the difference between the model collision term and the original binary collision term. The indicator is derived by controlling the quadratic terms in the modeling error with linear operators. Our numerical experiments show that such an error indicator is effective and computationally affordable.