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

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