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

具有移动边界的动力学方程的最小二乘近似无网格方法

• 批准号: 428845667
• 负责人: Professor Dr. Axel Klar
• 金额: --
• 依托单位: Dipartimento di Matematica e Informatica
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/428845667?language=en
• 关键词: Mesh; free; methods; least; squares

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.

这项建议涉及非线性动力学问题的数值方法的发展，如气体动力学的BGK方程，在移动几何形状的情况下。重点介绍了半拉格朗日方法和任意拉格朗日-欧拉方法的发展和研究。作为一种统一的特征，两种方法都使用了无网格移动最小二乘(MLS)近似。在半拉格朗日情形下，到目前为止，发展仅限于一维情形和低阶方法，而对于动力移动边界问题的ALE方法，则完全没有发展。因此，我们考虑将BGK方程的半拉格朗日移动边界方法推广到高维和高阶，以及高阶ALE方法的发展。这两种方法都需要函数或导数的重建过程。对于本方案，我们使用基于移动最小二乘近似的重建，这是因为它的简单性很好地适应于移动边界的问题。在项目过程中开发的方法将用于解决纳米技术的几个相关问题，并将作为测试案例。作为第一个例子，我们与S.Hardt，TU Darmstadt一起研究了温度梯度引起的流动中多组分气体的分离问题。作为第二个例子，我们考虑了由BGK方程给出的气相中二维运动的纳米粒子。此外，还研究了微电子机械系统中的流动问题。最后将这些算例与运动理论中经典的移动边界测试算例一起在三维中进行了讨论。