On the numerical signature of blow-up in hydrodynamic equations

TL;DR

This paper investigates numerical signatures of finite time blow-up in 2D complexified Euler equations, providing criteria to distinguish genuine blow-up from numerical artifacts in simulations.

Contribution

It introduces a geometrically consistent discretization method and identifies a growth rate signature of blow-up based on vorticity norms.

Findings

Growth rate of vorticity supremum norm indicates blow-up

Numerical stability verified across resolutions

Guidelines for validating blow-up in future simulations

Abstract

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.