Unconditional long-time stability of a velocity-vorticity method for the\n 2D Navier-Stokes equations

2015/11/25 by Heister, Timo, Maxim A. Olshanskii, Olshanskii, Maxim A. +2 · 3 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1511.08072

openalex publication_date 2015/11/25 · openalex created_date 2022/10/02 · openalex updated_date 2026/08/01

Abstract

We prove unconditional long-time stability for a particular\nvelocity-vorticity discretization of the 2D Navier-Stokes equations. The scheme\nbegins with a formulation that uses the Lamb vector to couple the usual\nvelocity-pressure system to the vorticity dynamics equation, and then\ndiscretizes with the finite element method in space and implicit-explicit BDF2\nin time, with the vorticity equation decoupling at each time step. We prove the\nmethod's vorticity and velocity are both long-time stable in the L² and\nH¹ norms, without any timestep restriction. Moreover, our analysis avoids\nthe use of Gronwall-type estimates, which leads us to stability bounds with\nonly polynomial (instead of exponential) dependence on the Reynolds number.\nNumerical experiments are given that demonstrate the effectiveness of the\nmethod.\n

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