Mesh-free methods with least squares approximations for kinetic equations with moving boundaries

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This proposal deals with the development of numerical methods for nonlinear kinetic problems, like the BGK-equation of gas kinetics, in situations with moving geometries. We concentrate on the development and investigation of Semi-Lagrangian and Arbitrary-Lagrangian-Eulerian (ALE) methods. As a unifying feature 'grid-free' Moving-Least-Squares (MLS) approximations are used for both approaches. In the Semi-Lagrangian case, up to now, the development has been restricted to 1D situations and low order methods, whereas in the case of ALE methods for kinetic moving boundary problems a development is completely missing. Thus, we consider the extension of Semi-Lagrangian moving boundary methods for the BGK-equation to higher dimensions and higher order and the development of higher order ALE methods. Both approaches require reconstruction procedures for functions or derivatives. For the present proposal we use a reconstruction based on a Moving-Least-Squares approximation, which is due to its simplicity well adapted for problems with moving boundaries. The methods developed in the course of the project will be used for the solution of several relevant problems from nano-technology, which will serve as test-cases. As a first example, we investigate, together with the group of S. Hardt, TU Darmstadt, separation problems for a multi-component gas in a flow induced by a temperature gradient. As a second example we consider 2-D moving nanoparticles in a gas phase given by the BGK equations. Moreover, flow in a Micro-Electro-Mechanical system is investigated. These examples are finally considered in 3-D together with a classical moving boundary test-case in kinetic theory.