Convergence of particle methods, particularly SPH

质点方法是数值模拟输运问题的一种有吸引力的现代方法。与有限差分、有限元和有限体积等经典方法不同，质点方法是基于问题的拉格朗日公式。因此，它们对于以平流为主的流动和具有不同拓扑结构的问题特别有用。近年来，在所有的粒子方法中，一种称为平滑粒子流体动力学(SPH)的技术变得非常流行。SPH是一种无网格、基于核的近似方法，已被广泛应用于天体物理、岩土工程、环境工程、水利工程、港口建设和固体物理等领域。尽管有一些尝试，但到目前为止，还没有对SPH方法进行严格的数学处理，特别是当它涉及到流体力学的应用时。因此，本项目的目标是对无粘流体流动的欧拉方程进行彻底的误差分析。为了尽可能灵活地选择时间离散化，我们首先只考虑SPH的空间离散化，并分析由此产生的半离散格式。之后，我们将分析一些显式的时间离散化方案。由于SPH有几个版本可以离散化空间导数，我们将检查和比较其中最流行的版本。最后，在第二步，我们将分析粘性流体的Navier-Stokes方程的SPH离散化。为此，到目前为止在这个项目中得到的误差估计必须扩展到也包括拉普拉斯算子的离散化。在所有这些情况下，我们感兴趣的是证明误差估计，特别强调不同离散化参数之间的联系。到目前为止，这些参数的选择仅依赖于问题和启发式。因此，离散化参数之间在数学上建立的精确联系也将在数值模拟方面带来显著改进。

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.