Generalized Gauss-Jacobi rules for discrete velocity method in Multiscale Flow Simulations

TL;DR

This paper introduces a novel generalized Gauss-Jacobi quadrature for the discrete velocity method, enhancing efficiency and accuracy across various flow regimes in multiscale gas flow simulations.

Contribution

The paper develops a new class of adjustable weight functions for quadrature, enabling flexible, adaptive discretization in DVM for multiscale flow simulations.

Findings

GGJQ outperforms traditional quadratures in accuracy and efficiency

Effective across continuum to rarefied flow regimes

Achieves superior computational performance in numerical tests

Abstract

The discrete velocity method (DVM) is a powerful framework for simulating gas flows across continuum to rarefied regimes, yet its efficiency remains limited by existing quadrature rules. Conventional infinite-domain quadratures, such as Gauss-Hermite, distribute velocity nodes globally and perform well near equilibrium but fail under strong nonequilibrium conditions. In contrast, finite-interval quadratures, such as Newton-Cotes, enable local refinement but lose efficiency near equilibrium. To overcome these limitations, we propose a generalized Gauss-Jacobi quadrature (GGJQ) for DVM, built upon a new class of adjustable weight functions. This framework systematically constructs one- to three-dimensional quadratures and maps the velocity space into polar or spherical coordinates, enabling flexible and adaptive discretization. The GGJQ accurately captures both near-equilibrium and highly rarefied regimes, as well as low- and high-Mach flows, achieving superior computational efficiency without compromising accuracy. Numerical experiments over a broad range of Knudsen numbers confirm that GGJQ consistently outperforms traditional Newton-Cotes and Gauss-Hermite schemes, offering a robust and efficient quadrature strategy for multiscale kinetic simulations.