Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods

TL;DR

This paper compares various numerical methods for simulating potential finite-time singularities in the 3D incompressible Euler equations, providing resolution estimates and analyzing their effectiveness using adaptive mesh refinement.

Contribution

It introduces a systematic comparison of spectral and finite difference methods for Euler equations, including adaptive mesh refinement and resolution scaling analysis.

Findings

Spectral methods with different dealiasing strategies perform variably.

Adaptive mesh refinement effectively captures singularity formation.

Resolution estimates are provided based on scaling behavior analysis.

Abstract

The numerical simulation of the 3D incompressible Euler equation is analyzed with respect to different integration methods. The numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods. Based on this comparison, a Kida-Pelz like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given. This estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective.