Adaptive Fluid Cohomology on Surfaces

2026-07-13

AI Summary

This work addresses the challenge of simulating inviscid incompressible fluid flow on non-simply connected surfaces, where existing methods struggle to simultaneously capture local and global flow features and often become unstable on low-quality meshes, corrupting harmonic components. The authors propose a spatiotemporally adaptive framework that employs posteriori error estimation to drive mesh refinement, integrates a Dormand–Prince 5(4) time-stepping scheme for enhanced temporal accuracy, and introduces a robust harmonic basis transfer algorithm to stably preserve harmonic components during mesh adaptation. This approach achieves, for the first time, adaptive conservation of harmonic fields in surface Euler equation simulations, maintaining high accuracy while reducing memory consumption by up to 86% and retaining numerical stability even on severely degraded meshes where conventional static methods fail.

Abstract

Simulating inviscid, incompressible fluids on non-simply-connected curved surfaces requires careful treatment of the flow's local and global behavior. While recent theoretical advancements have established the critical dynamics of the harmonic component in such flows, practical applications remain computationally restricted by a lack of spatial and temporal adaptivity. Furthermore, simulations on poor-quality meshes often lead to numerical instability and a failure to preserve the flow's underlying harmonic component when using naive interpolation methods. In this paper, we introduce Adaptive Fluid Cohomology, a framework that integrates dynamic spatial and temporal refinement into the simulation of the Euler equations. We leverage a posteriori error estimation to adjust spatial resolution on the fly, alongside a standard Dormand-Prince 5(4) time-stepping scheme for temporal accuracy. To ensure stability during mesh mutations, we develop a novel method that robustly transfers the harmonic basis during remeshing. While our experimental evaluation focuses on 2D surface flows, the underlying theoretical formulation is presented to capture the 3D setting as well. Our evaluation demonstrates that this adaptive approach accurately recreates the dynamics of high-resolution simulations while reducing the memory footprint by up to 86% and maintaining numerical stability even on poor-quality triangulations where static methods fail.

Problem

• adaptive simulation
• incompressible fluid
• harmonic component
• non-simply-connected surfaces
• numerical stability

Innovation

• adaptive simulation
• fluid cohomology
• harmonic basis transfer
• Euler equations on surfaces
• mesh refinement