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.

该提案介绍了用于非线性动力学问题的数值方法的发展，例如气体动力学的BGK-方程式在移动几何形状的情况下。我们专注于半拉格朗日和任意拉格朗日 - 欧拉（ALE）方法的开发和研究。作为一种统一的特征“无网格”移动最高方分（MLS）近似值用于两种方法。在半拉格朗日案件中，到目前为止，该开发仅限于1D情况和低阶方法，而对于动力学移动边界问题的啤酒方法，则完全缺少开发。因此，我们考虑将BGK方程式的半拉格朗日移动边界方法扩展到更高的维度和更高级别以及高阶啤酒方法的发展。两种方法都需要用于功能或衍生物的重建程序。对于本建议，我们使用基于移动最高方近似的重建，这是由于其简单性很好地适用于移动边界问题。项目过程中开发的方法将用于解决纳米技术中的几个相关问题，这些问题将用作测试案例。作为第一个例子，我们与S. hardt，Tu Darmstadt组一起研究了由温度梯度引起的流量中多组分气体的分离问题。作为第二个例子，我们考虑了BGK方程给出的气相中的2-D移动纳米颗粒。此外，研究了微机械系统中的流动。最终在3-D中考虑了这些示例，以及动力学理论中经典的运动边界测试案例。