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Convergence of particle methods, particularly SPH

粒子方法的收敛，特别是 SPH

基本信息

批准号：
262166066
负责人：
Professor Dr. Holger Wendland
金额：
--
依托单位：
Lehrstuhl Mathematik III - Angewandte und Numerische Analysis
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/262166066?language=en
关键词：
Convergence particle methods particularly SPH

项目摘要

Particle methods represent an attractive and modern method for the numerical simulation of transport problems. Particle methods are, in contrast to classical methods like Finite Differences, Finite Elements and Finite Volumes, based upon a Lagrangian formulation of the problem. Hence, they are particularly useful for advection dominated flows and for problems with a varying topology. In recent years, amongst all particle methods, a technique called SPH (smoothed particle hydrodynamics) has become extremely popular. SPH is a mesh-free, kernel-based approximation method, which is, for example, already employed in such diverse applications as astrophysics, geotechnical and environmental engineering, hydraulic engineering, harbour construction and solid-state physics. Despite some attempts, so far, there is no rigorous mathematical treatment of the SPH method, particularly when it comes to applications in fluid dynamics. Hence, it is the goal of this project to provide a thorough error analysis of the Euler equations for inviscid fluid flow. For the best possible flexibility regarding the choice of the time discretisation, we will first only look at the spacial discretisation with SPH and analyse the semi-discrete scheme resulting from this. After that, we will analyse some explicit time-discretisation schemes. Since there are several versions of SPH to discretise spatial derivatives, we will examine and compare the most popular ones amongst them. Finally, in a second step, we will analyse the SPH discretisation of the Navier-Stokes equations for viscous fluids. To this end, the error estimates derived thus far within this project have to be extended to also cover the discretisation of the Laplace operator. In all these cases, we are interested in proving error estimates with particular emphasis on the connection between the different discretisation parameters. So far, these parameters are only chosen problem dependent and heuristically. A precise mathematically established connection between the discretisation parameters will therefore also lead to a significant improvement when it comes to numerical simulation.

粒子方法是数值模拟输运问题的一种有吸引力的现代方法。粒子方法与经典方法如有限差分、有限元和有限差分相比，基于问题的拉格朗日公式。因此，他们是特别有用的平流占主导地位的流动和变化的拓扑结构的问题。近年来，在所有粒子方法中，一种称为SPH（平滑粒子流体动力学）的技术变得非常流行。SPH是一种无网格、基于核的近似方法，例如，它已经在天体物理学、岩土工程和环境工程、水利工程、港口建设和固体物理学等各种应用中得到了应用。尽管有一些尝试，但到目前为止，SPH方法还没有严格的数学处理，特别是在流体动力学中的应用。因此，这是本项目的目标，提供一个彻底的误差分析的欧拉方程的无粘流体流动。为了尽可能灵活地选择时间离散，我们将首先只考虑SPH的空间离散，并分析由此产生的半离散方案。在此之后，我们将分析一些显式时间离散格式。由于有几个版本的SPH离散空间导数，我们将检查和比较其中最流行的。最后，在第二步中，我们将分析粘性流体的Navier-Stokes方程的SPH离散。为此，迄今为止在这个项目中得出的误差估计必须扩展到也包括离散化的拉普拉斯算子。在所有这些情况下，我们有兴趣证明误差估计，特别强调不同的离散化参数之间的连接。到目前为止，这些参数只是选择问题相关和启发式。因此，在离散化参数之间建立精确的数学连接也将在数值模拟方面带来显着的改进。