论文标题
关于流体动力方程中爆炸的数值标志
On the numerical signature of blow-up in hydrodynamic equations
论文作者
论文摘要
流体动力部分微分方程中有限时间爆破的现象在分析和数学物理学中至关重要。尽管数值研究指导了理论突破，但确定观察到的计算结果是真实的还是仅仅是数值伪影。在这里，我们确定爆炸的数值签名。我们的研究基于在二维中的复杂欧拉方程，预计会立即爆破。通过几何一致的时空离散化，我们执行了几个数值实验并验证其计算稳定性。然后，我们根据涡度的最高规范的增长率随空间分辨率的增加而确定爆炸的签名。该研究的目的是在尚未解决分析的方程式中对可疑爆炸的未来数值实验进行交叉检查的指南。
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.
