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Öğe An Adaptive Time Filter Based Finite Element Method for the Velocity-Vorticity-Temperature Model of the Incompressible Non-Isothermal Fluid Flows(2020) Akbas, MineThis paper studies a velocity-vorticity-temperature (VVT) model of the Boussinesqequations and introduces a numerical method for solving that. The proposed numericalmethod adds separate three minimally intrusive steps, one for each fluid variable, exceptpressure, to the standard semi-implicit backward-Euler (BE) approximation of VVTmodel.The key idea in these intrusive steps is to post-process the BE approximatesolutions with 2-step, second order, linear time filters. The paper provides fullmathematical analysis of the proposed numerical method, and two numericalexperiments for that. The first numerical experiment verifies the predicted convergencerates while the second one shows the effectiveness of the method on a benchmarkproblem.Öğe Improving accuracy in the Leray model for incompressible nonisothermal flows via adaptive deconvolution-based nonlinear filtering(Wiley, 2021) Akbas, Mine; Bowers, AbigailThis paper considers a Leray regularization model of incompressible, nonisothermal fluid flows which uses nonlinear filtering based on indicator functions and introduces an efficient numerical method for solving it. The proposed method uses a multistep, second-order temporal discretization with a finite element (FE) spatial discretization in such a way that the resulting algorithm is linear at each time level and decouples the evolution equations from the velocity filter step. Since the indicator function chosen in this model is mathematically based on approximation theory, the proposed numerical algorithm can be analyzed robustly, i.e., the stability and convergence of the method is provable. A series of numerical tests are carried out to verify the theoretical convergence rates and to compare the algorithm with direct numerical simulation and the usual Leray-alpha model of the flow problem.Öğe Modular grad-div stabilization for the incompressible non-isothermal fluid flows(Elsevier Science Inc, 2021) Akbas, Mine; Rebholz, Leo G.This paper considers a modular grad-div stabilization method for approximating solutions of the time-dependent Boussinesq model of non-isothermal flows. The proposed method adds a minimally intrusive step to an existing Boussinesq code, with the key idea being that the penalization of the divergence errors, is only in the extra step (i.e. nothing is added to the original equations). The paper provides a full mathematical analysis by proving unconditional stability and optimal convergence of the methods considered. Numerical experiments confirm theoretical findings, and show that the algorithms have a similar positive effect as the usual grad-div stabilization. (C) 2020 Elsevier Inc. All rights reserved.Öğe A Numerical Study of a First Order Modular Grad-Div Stabilization for the Magnetohydrodynamics Equations(Amer Inst Physics, 2021) Akbas, MineThis paper proposes a stabilization method to approximate analytical solutions of magnetohydrodynamics (MHD) equations. The method adds two modular grad-div steps into fully-discrete finite element MHD solver. The main idea in these intrusive steps is to penalize the divergence of the velocity/magnetic fields both in L-2 and H-1-norms. The paper confirms the optimal convergence of the method, and gives numerical experiments which reveal positive effect of the method as in the usual grad-div stabilization.Öğe On the long-time stability of finite element solutions of the navier-stokes equations in a rotating frame of reference(2020) Akbas, MineThis paper studies the long-time stability behavior of the Navier-Stokes equations (NSE) in a rotating frame ofreference with atime accurate and adaptive finite element method. The proposed numerical scheme consists of twodecoupled steps. In the first step, the Navier-Stokes equations are solved with the standard linearized backwardEuler finite element method (BE-FEM). In the second step, the approximate velocity solution obtained in the firststep is post proceeded with a 2-step, linear time filter. It is proven that the approximate velocity solution is stablewith respect to??2-norm at all times. The novelty of the stability analysis is that the stability bound obtained for theapproximate velocity solution does not use any Gronwall-type estimate and is polynomially dependent on theReynolds number, which is not common in long-time stability notion. The paper also provides two numericalexperiments to test the algorithm. The first numerical experiment compares the ??2-norm of the velocity solutionof the proposed algorithmusing pressure-robust and non pressure-robustFE over longer time intervals. The resultsreveal that the scheme gives much more accurate velocity solutions with pressure-robust methods, especially forthe smaller ??. The second experiment, on the other hand, shows that the filter step increasesthe accuracy of theproposed numerical method over long-time intervals.Öğe Unconditional Long Time H(1)(-)Stability of a Velocity-Vorticity-Temperature Scheme for the 2D-Boussinesq System(Global Science Press, 2020) Akbas, MineThis paper proposes, analyzes and tests a velocity-vorticity-temperature (VVT) scheme for incompressible, non-isothermal fluid flow. VVT consists of complementing of the usual velocity-pressure-temperature system with the vorticity equation, coupling the systems through the convective terms. The proposed scheme uses BDF2LE time stepping, and a finite element spatial discretization. At each time step, the velocity-pressure equation, the vorticity equation and the temperature equation are all decoupled. A full analysis of the method is given that proves unconditional long-time H-1-stability, and shows the optimal convergence both in time and space. Theoretical convergence results are confirmed by a numerical test, and the effectiveness of the algorithm is revealed on a benchmark problem for Marsigli flow.
