The phenomenon of finite time blow-up in hydrodynamic partial differential equations is central in analysis and mathematical physics. While numerical studies have guided theoretical breakthroughs, it is challenging to determine if the observed computational results are genuine or mere numerical artifacts. Here we identify numerical signatures of blow-up. Our study is based on the complexified Euler equations in two dimensions, where instant blow-up is expected. Via a geometrically consistent spatiotemporal discretization, we perform several numerical experiments and verify their computational stability. We then identify a signature of blow-up based on the growth rates of the supremum norm of the vorticity with increasing spatial resolution. The study aims to be a guide for cross-checking the validity for future numerical experiments of suspected blow-up in equations where the analysis is not yet resolved.

翻译：流体动力学偏微分方程中的有限时间爆破现象是分析和数学物理的核心问题。尽管数值研究推动了理论突破，但确定观测到的计算结果究竟是真实现象还是数值伪影仍具挑战性。本文中，我们识别了爆破的数值特征。我们的研究基于二维复化欧拉方程——该方程预期会出现瞬时爆破。通过几何一致的时空离散化方法，我们开展了多项数值实验并验证了其计算稳定性。随后，我们基于涡量上确界范数随空间分辨率提高的增长速率，识别出一种爆破特征。本研究旨在为未来在理论尚未明确的方程中，对疑似爆破现象进行数值实验的有效性交叉验证提供指导。