Two-dimensional Gauss--Jacobi Quadrature for Multiscale Boltzmann Solvers

TL;DR

This paper introduces a two-dimensional Gauss-Jacobi quadrature scheme with a parameterized weight function and polar coordinate transformation, significantly improving velocity space discretization in multiscale Boltzmann solvers.

Contribution

It presents a novel quadrature method that enhances accuracy and efficiency in velocity space discretization for Boltzmann equations, addressing node mismatch issues.

Findings

Achieves up to 50x speed-up in highly rarefied conditions

Significantly improves accuracy over traditional methods

Reduces computational cost in multiscale gas flow simulations

Abstract

The discretization of velocity space plays a crucial role in the accuracy and efficiency of multiscale Boltzmann solvers. Conventional velocity space discretization methods suffer from uneven node distribution and mismatch issues, limiting the performance of numerical simulations. To address this, a Gaussian quadrature scheme with a parameterized weight function is proposed, combined with a polar coordinate transformation for flexible discretization of velocity space. This method effectively mitigates node mismatch problems encountered in traditional approaches. Numerical results demonstrate that the proposed scheme significantly improves accuracy while reducing computational cost. Under highly rarefied conditions, the proposed method achieves a speed-up of up to 50 times compared to the conventional Newton-Cotes quadrature, offering an efficient tool with broad applicability for numerical simulations of rarefied and multiscale gas flows.