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Analysis of second order and unconditionally stable BDF2‐AB2 method for the Navier‐Stokes equations with nonlinear time relaxation
Author(s) -
Isik Osman Rasit,
Yuksel Gamze,
Demir Bulent
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22276
Subject(s) - mathematics , nonlinear system , relaxation (psychology) , finite element method , order (exchange) , navier–stokes equations , compressibility , psychology , social psychology , physics , finance , quantum mechanics , economics , thermodynamics , engineering , aerospace engineering
In this study, we first consider a second order time stepping finite element BDF2‐AB2 method for the Navier‐Stokes equations (NSE). We prove that the method is unconditionally stable and O ( Δ t 2 ) accurate. Second, we consider a nonlinear time relaxation model which consists of adding a term “ κ | u − u ¯ | ( u − u ¯ ) ” to the Navier‐Stokes Equations with the algorithm depends on BDF2‐AB2 method. We prove that this method is unconditionally stable, too. We applied the BDF2‐AB2 method to several numeral experiments including flow around the cylinder. We have also applied BDF2‐AB2 method with nonlinear time relaxation to some problems. It is observed that when the equilibrium errors are high, applying BDF2‐AB2 with nonlinear time relaxation method to the problem yields lower equilibrium errors.