Vortex Methods for Incompressible Flows
Final Report Abstract
Significant progress was made during the course if this research project. The practical applicability of vortex methods, however, will require further efforts. The beginning of this project was marked by a great leap of progress in the problem of particle regularisation: together with Christian Rieger an efficient and stable method was developed, that constructs a globally defined function from point-values on a given point cloud. For now this method is limited to quasi-uniform point clouds; however it is also applicable in bounded domains. In these cases it solves one of the main problems of classical particle methods. Some interesting connections to kernel-based methods were observed; this lead to a co-operation with Paul Wilhelm and their application to the Vlasov-Poisson equation. Since I have left the academic world, in my spare time I work on the generalisation of this method to non-uniform particle clouds. The remainder of this project focused on the problem of computing the velocity field from a given vorticity field. The Biot–Savart law only solves this problem for the whole-space case, the solution in bounded domains turned out to be significantly more challenging. Together with Erick Schulz sharp existence, uniqueness, and regularity results were established, together with a numerical method that puts these results into practice. The complicated results that appear in this formulation can be solved analytically. The practical implementation of these analytic solutions was carried out together with Donat Weniger.
Publications
-
Analytic integration of the Newton potential over cuboids and an application to fast multipole methods. Journal of Numerical Mathematics, 30(2), 109-120.
Kirchhart, Matthias & Weniger, Donat
-
Discrete Projections: A Step Towards Particle Methods on Bounded Domains without Remeshing. SIAM Journal on Scientific Computing, 43(1), A609-A635.
Kirchhart, Matthias & Rieger, Christian
-
Div–curl problems and H1‐regular stream functions in 3D Lipschitz domains. Mathematical Methods in the Applied Sciences, 45(3), 1097-1117.
Kirchhart, Matthias & Schulz, Erick
-
An interpolating particle method for the Vlasov–Poisson equation. Journal of Computational Physics, 473, 111720.
Wilhelm, R. Paul & Kirchhart, Matthias
