2025/12/23 by Tim Brüers, Brüers, Tim, Christoph Lehrenfeld +3
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2512.20763
openalex publication_date 2025/12/23 · openalex created_date 2025/12/26 · openalex updated_date 2026/07/28
This paper presents a streamfunction-vorticity formulation for the Navier--Stokes and Euler equations on general surfaces. Notably, this includes non-simply connected surfaces, on which the harmonic components of the velocity field play a fundamental role in the dynamics. By relying only on scalar and finite-dimensional quantities, our formulation ensures that the resulting methods give exactly tangential and incompressible velocity fields, while also being pressure robust. Compared to traditional methods based on velocity-pressure formulations, where one can only guarantee these structural properties by increasing the computational costs, this is a key advantage. We rigorously validate our formulation by proving its equivalence to the well understood velocity-pressure formulation under reasonable regularity assumptions. Furthermore, we demonstrate the applicability of the approach with numerical examples.