曲面上的自适应流体上同调
Adaptive Fluid Cohomology on Surfaces
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中文总结 AI 辅助
研究在非单连通曲面上模拟无粘性、不可压缩流体的问题，提出自适应流体上同调框架，集成动态时空细化，利用后验误差估计和新方法保证稳定性，实验表明该方法能准确再现动力学，减少内存占用并保持稳定性。
中文摘要 AI 辅助
在非单连通曲面上模拟无粘性、不可压缩流体需要仔细处理流动的局部和全局行为。尽管近期理论进展已确定此类流动中谐波分量的关键动力学，但实际应用仍受缺乏时空适应性的计算限制。此外，使用简单插值方法时，在质量差的网格上进行模拟常导致数值不稳定且无法保留流动的基础谐波分量。本文引入自适应流体上同调框架，将动态时空细化集成到欧拉方程模拟中。利用后验误差估计即时调整空间分辨率，采用标准的多曼-普林斯5(4)时间步长方案保证时间精度。为确保网格变化时的稳定性，开发了一种在重新网格化期间稳健转移谐波基的新方法。实验评估聚焦二维表面流动，理论公式也适用于三维情况。评估表明，这种自适应方法能准确再现高分辨率模拟的动力学，同时将内存占用减少多达86%，即使在静态方法失败的质量差的三角剖分上也能保持数值稳定性。
英文摘要
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.
