Combining Hyperbolic Quadrature Method of Moments and Discrete-Velocity-Direction Models for Solving BGK-type Equations

Abstract

This paper introduces the discrete-velocity-direction model (DVDM) in conjunction with the hyperbolic quadrature method of moments (HyQMOM) to develop a multidimensional spatial-temporal approximation of the BGK equation, termed DVD-HyQMOM. Serving as a multidimensional extension of HyQMOM, DVD-HyQMOM model achieves higher accuracy than other DVDM submodels, especially with an increased number of abscissas. The efficiency and effectiveness of this model are demonstrated through various numerical tests.

Figures (9)

• Figure 1: 1-D Riemann problem with $\tau = 10^4$: profiles of density $\rho$, velocity $u$ and energy $E$ at $t = 0.2$. Orange, analytical solution. Blue, the DVD-EQMOM model. Green, the DVD-GQMOM model. Purple, the DVD-HyQMOM mode. Here the analytical result is the solution of the free-transport equation. In all models, we set $N = 8$ and the directions $\left\{\bm{l}_m = \left(\cos\frac{(2m-1)\pi}{16}, \sin\frac{(2m-1)\pi}{16}\right)^{T}\right\}^{8}_{m = 1}$. We set $n = 2$ for the DVD-HyQMOM and the DVD-GQMOM, and $M = 2$ for the DVD-EQMOM.
• Figure 2: 1-D Riemann problem with $\tau = 10^4$: profiles of density $\rho$, velocity $u$ and energy $E$ at $t = 0.2$. Orange, analytical solution. Blue, $n=2$. Green, $n = 3$. Purple, $n = 4$. Yellow, $n = 8$. In all cases, we set $N = 8$ and the directions $\left\{\bm{l}_m = \left(\cos\frac{(2m-1)\pi}{16}, \sin\frac{(2m-1)\pi}{16}\right)^{T}\right\}^{8}_{m = 1}$, using the DVD-HyQMOM model.
• Figure 3: 1-D Riemann problem with $\tau = 10^4$ at $t = 0.2$: (a) The correlation between the Mean Squared Errors of classical fluid quantities and $n$. Orange, $\rho$. Green, $E$. Blue, $u$. (b) The correlation between the elapsed time and $n$.
• Figure 4: 1-D Riemann problem with $\tau = 10^{-4}$: profiles of density $\rho$, velocity $u$ and energy $E$ at $t = 0.2$. Orange, analytical solution. Blue, the DVD-EQMOM model. Green, the DVD-GQMOM model. Purple, the DVD-HyQMOM model. Here the analytical result is the solution of the Euler equations xu151dana0. In all models, we set $N = 8$ and the directions $\left\{\bm{l}_m = \left(\cos\frac{(2m-1)\pi}{16}, \sin\frac{(2m-1)\pi}{16}\right)^{T}\right\}^{8}_{m = 1}$. We set $n = 2$ for the DVD-HyQMOM and the DVD-GQMOM, and $M = 2$ for the DVD-EQMOM.
• Figure 5: 1-D Riemann problem with $\tau = 10^{-4}$: profiles of density $\rho$, velocity $u$ and energy $E$ at $t = 0.2$. Orange, analytical solution. Blue, $n=2$. Green, $n = 3$. Purple, $n = 4$. In all cases, we set $N = 8$ and the directions $\left\{\bm{l}_m = \left(\cos\frac{(2m-1)\pi}{16}, \sin\frac{(2m-1)\pi}{16}\right)^{T}\right\}^{8}_{m = 1}$, using the DVD-HyQMOM model.