Vortex Methods for Incompressible Flows

在实践中发生的许多流动问题（如果不是大多数的话）都以湍流为特征。传统的基于网格的方法（如有限元）在应用于此类流动时面临着严重的问题：严格的时间步长限制、不稳定性或引入大量虚假数值粘度。虽然存在更先进的方案，但这种流动仍然是一个重大挑战。另一方面，粒子方法是基于方程对流部分的解析解，并且不受这些问题的困扰。特别是涡旋方法，具有许多理想的守恒特性。申请人最近的进展为在存在边界的情况下应用这些方法开辟了新的可能性。为复杂几何图形生成体积网格已被证明是一项劳动密集型、耗时的任务。几乎作为一个副产品，这些相同的结果也创造了进一步研究一类新的半解析的，无网格的求解泊松问题和有界域的热方程的机会。这些求解器只需要领域边界的网格，而不需要领域本身，这大大减轻了用户的负担。涡旋方法的特殊数学结构使我们可以把非线性流动方程看作是线性子问题的耦合。这些子问题和它们的解决方法允许一个数学上严格的分析和收敛性质可以建立。对于耦合方程，我们提出了几个数值测试用例作为基准。

Many if not most flow problems occurring in practice feature turbulence. Conventional, mesh-based methods like finite elements face severe problems when applied to such flows: stringent time-step constraints, instabilities, or the introduction of significant amounts of spurious numerical viscosity. While more advanced schemes do exist, such flows remain a significant challenge. Particle methods, on the other hand, are based on an analytical solution of the convective part of the equations and do not suffer from any of these problems. Vortex methods, in particular, feature many desirable conservation properties. Recent progress by the applicant opened new possibilities to apply these methods in the presence of boundaries. The generation of volumetric meshes for complicated geometries has proven to be a labour-intensive, time-consuming task. Almost as a side-product, these same results also created the opportunity of further research into a new class of semi-analytical, mesh-less solvers for the Poisson problem and the Heat equation in bounded domains. These solvers would only require a mesh of the domain's boundary instead of the domain itself, significantly reducing the burden on their users.The specific mathematical structure of vortex methods allows us to consider the non-linear flow equations as a coupling of linear sub-problems. These sub-problems and the methods applied for their solution allow for a mathematically rigorous analysis and convergence properties can be established. For the coupled equations we suggest several numerical test-cases as benchmarks.