Poisson quadrature method of moments for 2D kinetic equations with velocity of constant magnitude

Abstract

This work is concerned with kinetic equations with velocity of constant magnitude. We propose a quadrature method of moments based on the Poisson kernel, called Poisson-EQMOM. The derived moment closure systems are well defined for all physically relevant moments and the resultant approximations of the distribution function converge as the number of moments goes to infinity. The convergence makes our method stand out from most existing moment methods. Moreover, we devise a delicate moment inversion algorithm. As an application, the Vicsek model is studied for overdamped active particles. Then the Poisson-EQMOM is validated with a series of numerical tests including spatially homogeneous, one-dimensional and two-dimensional problems.

Figures (12)

• Figure 1: The Poisson kernels with different values of $r$.
• Figure 2: Reconstruction of the von Mises distribution by the Poisson-EQMOM.
• Figure 3: The spatially homogeneous case.
• Figure 4: The 1D Riemann problem (\ref{['eq:1D1']}) with a rarefaction wave. The profiles of density $\rho$ and velocity direction $\bar{\theta}$ at $t=4$. The 'macro' profiles are extracted from GAMBA2015.
• Figure 5: The 1D Riemann problem (\ref{['eq:1D2']}) with a shock wave. The profiles of density $\rho$ and velocity direction $\bar{\theta}$ at $t=4$. The 'macro' profiles are extracted from GAMBA2015.

Theorems & Definitions (13)

• Proposition 2.1
• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Lemma 2.6
• Proposition 2.2
• Proposition 2.3
• Proposition 3.1