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

TL;DR

This work introduces ATGJ, a two-dimensional Gaussian–Jacobi quadrature with a fully tunable weight in polar coordinates to discretize velocity space for multiscale Boltzmann solvers. By applying an arctangent-based mapping and a polar transform, ATGJ flexibly places nodes and weights to better capture non-equilibrium velocity distributions, reducing cost while improving accuracy. The method is validated on TDI cavity flow and hypersonic flow past a square cylinder, showing up to substantial speed-ups and accurate predictions with far fewer velocity points, and it recovers the Maxwellian limit as a special case. The framework offers robust performance across continuum and rarefied regimes and is extendable to higher dimensions with potential for automated parameter selection.

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.

Figures (6)

• Figure 1: Variations of bell-shaped weight functions under different parameter settings: (a) $\alpha = \frac{\pi}{2}\lambda + 2$ (b) $\alpha = \frac{\pi}{2}\lambda - 2$
• Figure 2: Illustration of flow geometries: (a) TDI cavity flow; (b) Hypersonic flow past a square cylinder.
• Figure 3: Temperature contours of the TDI cavity flow at $\mathit{Kn} = 0.001$: (a) $\lambda = 5$; (b) $\lambda = 500$. The white solid lines denote the analytical solution, and the black dashed lines correspond to the ATGJ results.
• Figure 4: Temperature field and streamlines of the TDI cavity flow at Kn = 10: (a) NC rule with 201 × 201 nodes; (b) ATGJ rule with 8 × 90 nodes.
• Figure 5: Temperature and velocity profiles along the horizontal and vertical centerlines for TDI cavity flow.