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

TL;DR

This paper addresses the efficiency-accuracy trade-off in velocity-space quadrature for the discrete velocity method in multiscale gas dynamics by introducing Generalized Gauss–Jacobi quadrature (GGJQ) with an adjustable weight function and radial/polar/spherical mappings. The approach constructs 1D–3D quadratures that concentrate nodes where the distribution function is most significant, seamlessly transitioning from near-equilibrium to highly rarefied regimes. The GGJQ is analyzed theoretically and validated against six benchmark problems (shock tube, shock structure, thermally driven cavity, supersonic cylinder, lid-driven cavity, and spherical Fourier flow), showing accuracy comparable to traditional rules with orders-of-magnitude reductions in node counts. The method promises substantial practical impact by enabling robust, efficient deterministic simulations across continuum to free-m molecular flows in MEMS, aerospace, and related multiscale applications.

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.

Figures (20)

• Figure 1: Bell-shaped surfaces and their projections of 2D weight functions. (a)$\beta =\alpha +10$; (b)$\beta =\alpha -10$.
• Figure 2: Illustration of discrete velocity distribution for 1D shock tube.
• Figure 3: Density, velocity, and temperature profiles of the 1D shock tube test ($\mu_0=10$).
• Figure 4: Density, velocity, and temperature profiles of the 1D shock tube test ($\mu_0=1$).
• Figure 5: Density, velocity, and temperature profiles of the 1D shock tube test ($\mu_0=10^{-3}$).