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Moving boundary problems for the BGK model of rarefied gas dynamics

A new semilagrangian method is presented for the numerical solution of the BGK model of the Boltzmann equation in a domain with moving boundary.
The method is based on discretization of the equation on a fixed grid in space and velocity. The equation is discretized in characteristic form, and the distribution function is reconstructed at the foot of the characteristics by a third order piecewise Hermite interpolation. Reflecting moving boundary at the piston are suitably described by assigning the value of the distribution function at ghost cells. A comparison with Euler equation of gas dynamics
for the piston problem has been performed in the case of small Knudsen number.
This work is motivated by the computation of rarefied
ow in MEMS (Micro Electro Mechanical Systems) [3]. The size of such devices is small enough that gas ow require a kinetic treatment even at normal pressure and temperature conditions. Micro accelerators are often composed of several
elements, each of which consists of a moving part, the shuttle, which is free to oscillate inside a fixed part, the stator. Although under certain conditions one can obtain an accurate description of the ow by quasi-static approximation
[2], more general flow conditions inside the element requires the treatment of a domain whose boundaries are not xed. As a warm up problem, we consider the evolution of a gas in a one dimensional piston. Since we are interested in description of the moving boundary, we choose the simple BGK model to describe the gas. The numerical method that we use is a deterministic semilagrangian method on a fixed grid in space and velocity. Such a method is illustrated in detail in paper [5].
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