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arXiv · 2512.20763

Streamfunction-vorticity formulation for incompressible viscid and inviscid flows on general surfaces

Tim Brüers·Christoph Lehrenfeld·Max Wardetzky
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Abstract

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.

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Tim Brüers, Christoph Lehrenfeld, Max Wardetzky. 2025-12-23. Streamfunction-vorticity formulation for incompressible viscid and inviscid flows on general surfaces. https://arxiv.org/abs/2512.20763

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