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.

该提案涉及在几何形状移动的情况下开发非线性动力学问题的数值方法，例如气体动力学的 BGK 方程。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。作为统一特征，“无网格”移动最小二乘法 (MLS) 近似被用于这两种方法。到目前为止，在半拉格朗日情况下，其发展仅限于一维情况和低阶方法，而在用于动力学移动边界问题的 ALE 方法中，完全缺乏发展。因此，我们考虑将 BGK 方程的半拉格朗日移动边界方法扩展到更高维度和更高阶，以及开发更高阶 ALE 方法。两种方法都需要函数或导数的重构过程。对于当前的建议，我们使用基于移动最小二乘近似的重建，这是由于其简单性非常适合移动边界的问题。该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，并将作为测试用例。作为第一个例子，我们与达姆施塔特工业大学的 S. Hardt 团队一起研究了由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，我们考虑由 BGK 方程给出的气相中的二维移动纳米颗粒。此外，还研究了微机电系统中的流动。这些例子最终与动力学理论中的经典移动边界测试用例一起以 3D 形式进行考虑。