Discrete-velocity-direction models of BGK-type with minimum entropy: I. Basic idea

TL;DR

This paper introduces a novel discrete-velocity-direction model for rarefied flow simulation, utilizing entropy minimization to determine equilibrium states, and demonstrates its basic properties and computational advantages.

Contribution

It proposes a new discrete-velocity-direction model with entropy-based equilibrium, enabling efficient computation and extension to multidimensional problems.

Findings

Discrete equilibriums can be found via convex optimization.

The model satisfies the H-theorem.

Numerical experiments validate the model's effectiveness.

Abstract

In this series of works, we develop a discrete-velocity-direction model (DVDM) with collisions of BGK-type for simulating rarefied flows. Unlike the conventional kinetic models (both BGK and discrete-velocity models), the new model restricts the transport to finite fixed directions but leaves the transport speed to be a 1-D continuous variable. Analogous to the BGK equation, the discrete equilibriums of the model are determined by minimizing a discrete entropy. In this first paper, we introduce the DVDM and investigate its basic properties, including the existence of the discrete equilibriums and the $H$-theorem. We also show that the discrete equilibriums can be efficiently obtained by solving a convex optimization problem. The proposed model provides a new way in choosing discrete velocities for the computational practice of the conventional discrete-velocity methodology. It also facilitates a convenient multidimensional extension of the extended quadrature method of moments. We validate the model with numerical experiments for two benchmark problems at moderate computational costs.