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On Convergence and Performance of Iterative Methods with FourthOrder Compact Schemes
 Numer. Methods Partial Differential Equations
, 1998
"... We study the convergence and performance of iterative methods with the fourthorder compact discretization schemes for the one and two dimensional convectiondiffusion equations. For the one dimensional problem, we investigate the symmetrizability of the coefficient matrix and derive analytical formul ..."
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Cited by 25 (16 self)
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We study the convergence and performance of iterative methods with the fourthorder compact discretization schemes for the one and two dimensional convectiondiffusion equations. For the one dimensional problem, we investigate the symmetrizability of the coefficient matrix and derive analytical formula for the spectral radius of the point Jacobi iteration matrix. For the two dimensional problem, we conduct Fourier analysis to determine the error reduction factors of several basic iterative methods and comment on their potential use as the smoothers for the multilevel methods. Finally, we perform numerical experiments to verify our Fourier analysis results. Key words: Convectiondiffusion equation, iterative methods, fourthorder compact discretization schemes. 1 Introduction We first consider the one dimensional (1D) convectiondiffusion equation \Gammau xx (x) + p(x)u x (x) = f(x); x 2 (a; b); u(a) = g 1 ; u(b) = g 2 : (1) This equation often appears in the description of transport ...
Multigrid Acceleration Techniques and Applications to the Numerical Solution of Partial Differential Equations
, 1997
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Fast and High Accuracy Multigrid Solution of the Three Dimensional Poisson Equation
 J. Comput. Phys
, 1998
"... We employ a fourthorder compact finite difference scheme (FOS) with the multigrid algorithm to solve the three dimensional Poisson equation. We test the influence of different orderings of the grid space and different gridtransfer operators on the convergence and efficiency of our high accuracy al ..."
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Cited by 18 (7 self)
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We employ a fourthorder compact finite difference scheme (FOS) with the multigrid algorithm to solve the three dimensional Poisson equation. We test the influence of different orderings of the grid space and different gridtransfer operators on the convergence and efficiency of our high accuracy algorithm. Fourier smoothing analysis is conducted to show that FOS has a smaller smoothing factor than the traditional secondorder central difference scheme (CDS). A new method of Fourier smoothing analysis is proposed for the partially decoupled redblack GaussSeidel relaxation with FOS. Numerical results are given to compare the computed accuracy and the computational efficiency of FOS with multigrid against CDS with multigrid. 1991 Mathematical Subject Classification: 65F10, 65N06, 65N22, 65N55. Key words: Poisson equation, multigrid method, fourthorder compact scheme, Fourier smoothing analysis. 1 Introduction Numerical simulation of threedimensional (3D) problems tends to be comput...
High Accuracy Multigrid Solution of the 3D ConvectionDiffusion Equation
 Appl. Math. Comput
, 1998
"... We present an explicit fourthorder compact finite difference scheme for approximating the three dimensional convectiondiffusion equation with variable coefficients. This 19point formula is defined on a uniform cubic grid. Fourier smoothing analysis is performed to show that the smoothing factor o ..."
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Cited by 16 (4 self)
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We present an explicit fourthorder compact finite difference scheme for approximating the three dimensional convectiondiffusion equation with variable coefficients. This 19point formula is defined on a uniform cubic grid. Fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1. We design a parallelizationoriented multigrid method for fast solution of the resulting linear system using a fourcolor GaussSeidel relaxation technique for robustness and efficiency, and a scaled residual injection operator to reduce the cost of multigrid intergrid transfer operator. Numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method. Several test problems are solved and highly accurate solutions of the 3D convectiondiffusion equations are ob...
Fast multilevel methods for Markov chains
"... This paper describes multilevel methods for the calculation of the stationary probability vector of large, sparse, irreducible Markov chains. In particular, several recently proposed significant improvements to the multilevel aggregation method of Horton and Leutenegger are described and compared. F ..."
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Cited by 5 (2 self)
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This paper describes multilevel methods for the calculation of the stationary probability vector of large, sparse, irreducible Markov chains. In particular, several recently proposed significant improvements to the multilevel aggregation method of Horton and Leutenegger are described and compared. Furthermore, we propose a very simple improvement of that method using an overcorrection mechanism. We also compare with more traditional iterative methods for Markov chains such as weighted Jacobi, twolevel aggregation/disaggregation, and preconditioned stabilized biconjugate gradient and generalized minimal residual method. Numerical experiments confirm that our improvements lead to significant speedup, and result in multilevel methods that are competitive with leading iterative solvers for Markov chains. Copyright © 2011
Numerical Simulation of 2D Square Driven Cavity Using Fourth Order Compact Finite Difference Schemes
, 2000
"... Fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large Reynolds numbers. The governing NavierStokes equation is linearized in streamfunction and vorticity formulation. The fourth order co ..."
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Cited by 3 (2 self)
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Fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large Reynolds numbers. The governing NavierStokes equation is linearized in streamfunction and vorticity formulation. The fourth order compact approximation schemes are coupled with fourth order approximations for velocities and vorticity boundaries. Numerical solutions are obtained for square driven cavity flow at high Reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods. Key words: Incompressible NavierStokes equations, square driven cavity problem, convection diffusion equation, multigrid method, fourth order compact discretization. Mathematics Subject Classification: 65N06, 65N22, 65F10. 1 Introduction The NavierStokes equations have been used to model fluid dynamics phenomena describing flows of an incompressible viscous fluid. These equations are high...
Optimal Injection Operator and High Order Schemes for Multigrid Solution of 3D Poisson Equation
, 1999
"... We present a multigrid solution of the three dimensional Poisson equation with a fourth order 19point compact finite difference scheme. Using a redblack ordering of the grid points and some geometric considerations, we derive an optimal scaled injection operator for the multigrid algorithm. Num ..."
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Cited by 1 (0 self)
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We present a multigrid solution of the three dimensional Poisson equation with a fourth order 19point compact finite difference scheme. Using a redblack ordering of the grid points and some geometric considerations, we derive an optimal scaled injection operator for the multigrid algorithm. Numerical computations show that this operator yields not only the smallest overall CPU time, but also the best convergence rate compared to other more traditional projection operators. In addition, we present a family of 19point compact schemes and numerically show that each one has a different optimal scaled injection operator. 1991 Mathematical Subject Classification : 65F10, 65N06, 65N22, 65N55. Key words: 3D Poisson equation, fourth order compact discretization, multigrid method, scaled injection operator. 1 Introduction In [4], we compared multigrid solution methods using a fourth order (9point) and a second order (5point) finite difference approximations of the two dimensional...
MultiLevel Minimal Residual Smoothing: A Family of General Purpose Multigrid Acceleration Techniques
, 1998
"... We employ multilevel minimal residual smoothing (MRS) as a preoptimization technique to accelerate standard multigrid convergence. The MRS method is used to improve the current multigrid iterate by smoothing its corresponding residual before the latter is projected to the coarse grid. We develop d ..."
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We employ multilevel minimal residual smoothing (MRS) as a preoptimization technique to accelerate standard multigrid convergence. The MRS method is used to improve the current multigrid iterate by smoothing its corresponding residual before the latter is projected to the coarse grid. We develop different schemes for implementing MRS technique on the finest grid and on the coarse grids, and several versions of the inexact MRS technique. Numerical experiments are conducted to show the efficiency of the multilevel and inexact MRS techniques. Key words: Minimal residual smoothing, multigrid method, residual scaling techniques. AMS subject classifications: 65F10, 65N06. 1 Introduction We propose a family of multilevel minimal residual smoothing (MRS) techniques as preoptimization acceleration schemes to speed up the convergence of the standard multigrid method for solving large sparse linear system A h u h = f h : (1) Eq. (1) usually results from discretized partial differenti...
Optimal Injection Operator and High Order Schemes for Multigrid Solution of 3D Poisson Equation
, 1999
"... Abstract We present a multigrid solution of the three dimensional Poisson equation with a fourth order 19point compact finite difference scheme. Using a redblack ordering of the grid points and some geometric considerations, we derive an optimal scaled injection operator for the multigrid algorith ..."
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Abstract We present a multigrid solution of the three dimensional Poisson equation with a fourth order 19point compact finite difference scheme. Using a redblack ordering of the grid points and some geometric considerations, we derive an optimal scaled injection operator for the multigrid algorithm. Numerical computations show that this operator yields not only the smallest overall CPU time, but also the best convergence rate compared to other more traditional projection operators. In addition, we present a family of 19point compact schemes and numerically show that each one has a different optimal scaled injection operator. 1991 Mathematical Subject Classification: 65F10, 65N06, 65N22, 65N55. Key words: 3D Poisson equation, fourth order compact discretization, multigrid method, scaled injection operator.
Numerical Simulation of 2D Square Driven Cavity Using Fourth Order Compact Finite Difference Schemes \Lambda
, 2000
"... Abstract Fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large Reynolds numbers. The governing NavierStokes equation is linearized in streamfunction and vorticity formulation. The fourth or ..."
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Abstract Fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large Reynolds numbers. The governing NavierStokes equation is linearized in streamfunction and vorticity formulation. The fourth order compact approximation schemes are coupled with fourth order approximations for velocities and vorticity boundaries. Numerical solutions are obtained for square driven cavity flow at high Reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods. Key words: Incompressible NavierStokes equations, square driven cavity problem, convection diffusion equation, multigrid method, fourth order compact discretization.