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On the numerical signature of blow-up in hydrodynamic equations

Erik Jansson, Klas Modin

TL;DR

The paper addresses the challenge of confirming finite-time blow-up in hydrodynamic PDEs by studying the complexified 2-D Euler equations on the sphere, a setting where instant blow-up is expected. It develops a geometrically faithful Zeitlin discretization via the Hoppe–Yau construction to produce a finite-dimensional, isospectral matrix analogue that preserves key invariants and structure. Through careful numerical experiments, the authors demonstrate computational stability of the observed blow-up (casimirs are preserved up to round-off, reversibility holds, and step-size changes do not artifactually induce blow-up) and reveal a clear resolution-dependent signature: the blow-up time decreases with increasing spatial resolution and the growth rate scales with $N$, suggesting ill-posedness in the infinite-dimensional limit. The proposed practical criterion—monitoring growth rates of $\|W\|$ across resolutions—provides a robust diagnostic for distinguishing true blow-up from numerical artifacts in equations where theory remains unresolved.

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.

On the numerical signature of blow-up in hydrodynamic equations

TL;DR

The paper addresses the challenge of confirming finite-time blow-up in hydrodynamic PDEs by studying the complexified 2-D Euler equations on the sphere, a setting where instant blow-up is expected. It develops a geometrically faithful Zeitlin discretization via the Hoppe–Yau construction to produce a finite-dimensional, isospectral matrix analogue that preserves key invariants and structure. Through careful numerical experiments, the authors demonstrate computational stability of the observed blow-up (casimirs are preserved up to round-off, reversibility holds, and step-size changes do not artifactually induce blow-up) and reveal a clear resolution-dependent signature: the blow-up time decreases with increasing spatial resolution and the growth rate scales with , suggesting ill-posedness in the infinite-dimensional limit. The proposed practical criterion—monitoring growth rates of across resolutions—provides a robust diagnostic for distinguishing true blow-up from numerical artifacts in equations where theory remains unresolved.

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.
Paper Structure (10 sections, 1 theorem, 45 equations, 9 figures)

This paper contains 10 sections, 1 theorem, 45 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Geometric preliminaries
  3. The complexified Euler equations
  4. The complexified vorticity equation
  5. Properties of the complexified Euler equations
  6. Zeitlin's model for the complexified Euler equations
  7. Matrix vorticity equations
  8. Numerical study of blow-up
  9. Experiment 1: The observed blow-up is computationally stable
  10. Experiment 2: Time to blow-up decreases as spatial resolution increases

Key Result

Proposition 3.1

The vorticity formulation of eq:compeuler is,

Figures (9)

  • Figure 1: The skew-Hermitian part (real component) of the vorticity matrix. Note that, while the component is initially small, it grows rapidly and reaches numerical blow-up at $t = 0.04619$.
  • Figure 2: The Hermitian part (imaginary component) of the vorticity matrix. It grows rapidly and reaches numerical blow-up at $t = 0.04619$.
  • Figure 3: The growth in time of the spectral norm for the $N=256$ solution displayed in \ref{['fig:imag_forward', 'fig:real_forward']}, together with the corresponding growths for solutions with $N = 64, 128, 512$.
  • Figure 4: The skew-Hermitian part (real component) of the vorticity matrix in the reversed simulation. Note that, while the component is initially at the threshold of blow-up at $T = 0$, it rapidly decreases and has returned to the initial conditions of the first simulation at $T = 0.04619$.
  • Figure 5: The Hermitian part (imaginary component) of the vorticity matrix. Note that, while the component is initially at the threshold of blow-up at $T = 0$, it decreases rapidly and has at $T = 0.04619$ returned to the initial conditions of the first simulation.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Proposition 3.1
  • proof
  • Remark 4.1