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.

这个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，在移动几何的情况下。我们专注于发展和研究半拉格朗日和拉格朗日欧拉（ALE）方法。作为一个统一的功能“网格自由”移动最小二乘（MLS）近似用于这两种方法。在半拉格朗日的情况下，到目前为止，发展已被限制在一维的情况下，低阶的方法，而在ALE方法的动力学移动边界问题的发展是完全缺失的。因此，我们考虑BGK-方程的半拉格朗日移动边界方法向高维和高阶的扩展以及高阶ALE方法的发展。这两种方法都需要函数或导数的重建过程。对于目前的建议，我们使用的重建基于移动最小二乘近似，这是由于其简单性，以及适合于移动边界的问题。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，这些问题将作为测试案例。作为第一个例子，我们研究，与S。Hardt，TU达姆施塔特，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2-D移动的纳米粒子在气相中给出的BGK方程。此外，在微机电系统中的流动进行了研究。这些例子最后被认为是在3-D连同一个经典的动边界的测试情况下，在动力学理论。