• 제목, 요약, 키워드: Uzawa method

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OPTIMAL ERROR ESTIMATE FOR SEMI-DISCRETE GAUGE-UZAWA METHOD FOR THE NAVIER-STOKES EQUATIONS

  • Pyo, Jae-Hong
    • 대한수학회보
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    • v.46 no.4
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    • pp.627-644
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    • 2009
  • The gauge-Uzawa method which has been constructed in [11] is a projection type method to solve the evolution Navier-Stokes equations. The method overcomes many shortcomings of projection methods and displays superior numerical performance [11, 12, 15, 16]. However, we have obtained only suboptimal accuracy via the energy estimate in [11]. In this paper, we study semi-discrete gauge-Uzawa method to prove optimal accuracy via energy estimate. The main key in this proof is to construct the intermediate equation which is formed to gauge-Uzawa algorithm. We will estimate velocity errors via comparing with the intermediate equation and then evaluate pressure errors via subtracting gauge-Uzawa algorithm from Navier-Stokes equations.

SEMI-CONVERGENCE OF THE PARAMETERIZED INEXACT UZAWA METHOD FOR SINGULAR SADDLE POINT PROBLEMS

  • YUN, JAE HEON
    • 대한수학회보
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    • v.52 no.5
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    • pp.1669-1681
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    • 2015
  • In this paper, we provide semi-convergence results of the parameterized inexact Uzawa method with singular preconditioners for solving singular saddle point problems. We also provide numerical experiments to examine the effectiveness of the parameterized inexact Uzawa method with singular preconditioners.

A CLASSIFICATION OF THE SECOND ORDER PROJECTION METHODS TO SOLVE THE NAVIER-STOKES EQUATIONS

  • Pyo, Jae-Hong
    • Korean Journal of Mathematics
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    • v.22 no.4
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    • pp.645-658
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    • 2014
  • Many projection methods have been progressively constructed to find more accurate and efficient solution of the Navier-Stokes equations. In this paper, we consider most recently constructed projection methods: the pressure correction method, the gauge method, the consistent splitting method, the Gauge-Uzawa method, and the stabilized Gauge-Uzawa method. Each method has different background and theoretical proof. We prove equivalentness of the pressure correction method and the stabilized Gauge-Uzawa method. Also we will obtain that the Gauge-Uzawa method is equivalent to the gauge method and the consistent splitting method. We gather theoretical results of them and conclude that the results are also valid on other equivalent methods.

AN OVERVIEW OF BDF2 GAUGE-UZAWA METHODS FOR INCOMPRESSIBLE FLOWS

  • Pyo, Jae-Hong
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.15 no.3
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    • pp.233-251
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    • 2011
  • The Gauge-Uzawa method [GUM] in [9] which is a projection type algorithm to solve evolution Navier-Stokes equations has many advantages and superior performance. But this method has been studied for backward Euler time discrete scheme which is the first order technique, because the classical second order GUM requests rather strong stability condition. Recently, the second order time discrete GUM was modified to be unconditionally stable and estimated errors in [12]. In this paper, we contemplate several GUMs which can be derived by the same manner within [12], and we dig out properties of them for both stability and accuracy. In addition, we evaluate an stability condition for the classical GUM to construct an adaptive GUM for time to make free from strong stability condition of the classical GUM.

THE STABILITY OF GAUGE-UZAWA METHOD TO SOLVE NANOFLUID

  • JANG, DEOK-KYU;KIM, TAEK-CHEOL;PYO, JAE-HONG
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.24 no.2
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    • pp.121-141
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    • 2020
  • Nanofluids is the fluids mixed with nanoscale particles and the mixed nano size materials affect heat transport. Researchers in this field has been focused on modeling and numerical computation by engineers In this paper, we analyze stability constraint of the dominant equations and check validate of the condition for most kinds of materials. So we mathematically analyze stability of the system. Also we apply Gauge-Uzawa algorithm to solve the system and prove stability of the method.

ERROR ESTIMATES FOR THE FULLY DISCRETE STABILIZED GAUGE-UZAWA METHOD -PART I: THE NAVIER-STOKES EQUATIONS

  • Pyo, Jae-Hong
    • Korean Journal of Mathematics
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    • v.21 no.2
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    • pp.125-150
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    • 2013
  • The stabilized Gauge-Uzawa method (SGUM), which is a second order projection type algorithm to solve the time-dependent Navier-Stokes equations, has been newly constructed in 2013 Pyo's paper. The accuracy of SGUM has been proved only for time discrete scheme in the same paper, but it is crucial to study for fully discrete scheme, because the numerical errors depend on discretizations for both space and time, and because discrete spaces between velocity and pressure can not be chosen arbitrary. In this paper, we find out properties of the fully discrete SGUM and estimate its errors and stability to solve the evolution Navier-Stokes equations. The main difficulty in this estimation arises from losing some cancellation laws due to failing divergence free condition of the discrete velocity function. This result will be extended to Boussinesq equations in the continuous research (part II) and is essential in the study of part II.

THE SECOND-ORDER STABILIZED GAUGE-UZAWA METHOD FOR INCOMPRESSIBLE FLOWS WITH VARIABLE DENSITY

  • Kim, Taek-cheol;Pyo, Jae-Hong
    • Korean Journal of Mathematics
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    • v.27 no.1
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    • pp.193-219
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    • 2019
  • The Navier-Stokes equations with variable density are challenging problems in numerical analysis community. We recently built the 2nd order stabilized Gauge-Uzawa method [SGUM] to solve the Navier-Stokes equations with constant density and have estimated theoretically optimal accuracy. Also we proved that SGUM is unconditionally stable. In this paper, we apply SGUM to the Navier-Stokes equations with nonconstant variable density and find out the stability condition of the algorithms. Because the condition is rather strong to apply to real problems, we consider Allen-Cahn scheme to construct unconditionally stable scheme.

ACCELERATION OF ONE-PARAMETER RELAXATION METHODS FOR SINGULAR SADDLE POINT PROBLEMS

  • Yun, Jae Heon
    • 대한수학회지
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    • v.53 no.3
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    • pp.691-707
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    • 2016
  • In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR methods with scaled preconditioners by comparing their performance with the parameterized Uzawa method with optimal parameters.