Vortex Methods for Incompressible Flows

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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.