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An Explicit FourthOrder Compact Finite Difference Scheme for Three Dimensional ConvectionDiffusion Equation
, 1997
"... 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. We compare advantages and implementation cost of the new scheme with the st ..."
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Cited by 24 (10 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. We compare advantages and implementation cost of the new scheme with the standard 7point scheme in the context of basic iterative methods. Numerical examples are used to verify the fourthorder convergence rate of the scheme and to show that the GaussSeidel iterative method converges for large values of the convection coefficients. Some algebraic properties of the coefficient matrices arising from different discretization schemes are compared. We also comment on potential use of the fourthorder compact scheme with multilevel iterative methods.
High Accuracy Stable Numerical Solution of 1D Microscale Heat Transport Equation
, 2000
"... We investigate the use of a fourth order compact finite difference scheme for solving an one dimensional heat transport equation at the microscale. The fourth order compact scheme is used with a CrankNicholson type integrator by introducing an intermediate function for the heat transport equation. ..."
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Cited by 6 (3 self)
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We investigate the use of a fourth order compact finite difference scheme for solving an one dimensional heat transport equation at the microscale. The fourth order compact scheme is used with a CrankNicholson type integrator by introducing an intermediate function for the heat transport equation. The new scheme is proved to be unconditionally stable with respect to initial values. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second order spatial discretization. It is shown that the new scheme is computationally more efficient and more accurate than the second order scheme. Key words: Heat transport equation, finite difference, fourth order compact scheme, CrankNicholson integrator. Mathematics Subject Classification: 65M06, 65N12. Technical Report 29700, Department of Computer Science, University of Kentucky, Lexington, KY, 2000. y Email: jzhang@cs.uky.edu. URL: jzhang@cs.uky.edu/jzhang. The research of this author was support...
Multigrid Method and Fourth Order Compact Difference Scheme for 2D Poisson Equation with Unequal Meshsize Discretization
, 2001
"... A fourth order compact difference scheme with unequal meshsizes in different coordinate directions is employed to discretize two dimensional Poisson equation in a rectangular domain. Multigrid methods using a partial semicoarsening strategy and line GaussSeidel relaxation are designed to solve t ..."
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Cited by 5 (1 self)
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A fourth order compact difference scheme with unequal meshsizes in different coordinate directions is employed to discretize two dimensional Poisson equation in a rectangular domain. Multigrid methods using a partial semicoarsening strategy and line GaussSeidel relaxation are designed to solve the resulting sparse linear systems. Numerical experiments are conducted to test accuracy of the fourth order compact difference scheme and to compare it with the standard second order difference scheme. Convergence behavior of the partial semicoarsening and line GaussSeidel relaxation multigrid methods is examined experimentally. Key words: Poisson equation, fourth order compact scheme, unequal meshsize, multigrid method, semicoarsening. Mathematics Subject Classification: 65F10, 65N06, 65N22, 65N55, 76D07. 1 Introduction We are interested in the high accuracy numerical solution of two dimensional (2D) Poisson equation of the form u xx (x; y) + u yy (x; y) = f(x; y); (x; y) 2\Omeg...
Fourth Order Compact Difference Scheme for 3D Convection Diffusion Equation with Boundary Layers on Nonuniform Grids
, 2000
"... We present a fourth order compact finite difference scheme for a general three dimensional convection diffusion equation with variable coefficients on a uniform cubic grid. This high order compact difference scheme is used to solve convection diffusion equation with boundary layers on a three dimens ..."
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Cited by 3 (1 self)
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We present a fourth order compact finite difference scheme for a general three dimensional convection diffusion equation with variable coefficients on a uniform cubic grid. This high order compact difference scheme is used to solve convection diffusion equation with boundary layers on a three dimensional nonuniform grid. We compare the computed accuracy and computational efficiency of the fourth order compact difference scheme with that of the standard central difference scheme and the first order upwind difference scheme. Several convection diffusion problems are solved numerically to validate the proposed fourth order compact scheme. Key words: convection diffusion equation, boundary layer, grid stretching, fourth order Technical Report 29800, Department of Computer Science, University of Kentucky, Lexington, KY, 2000. y Email: jzhang@cs.uky.edu, URL: http://www.cs.uky.edu/jzhang. The research of this author was supported in part by the U.S. National Science Foundation under g...
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...
High Order Compact Difference Scheme and Multigrid Method for 2D Elliptic Problems with Variable Coefficients and Interior/Boundary Layers on Nonuniform Grids
, 2015
"... In this paper, a high order compact difference scheme and a multigrid method are proposed for solving twodimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a non ..."
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In this paper, a high order compact difference scheme and a multigrid method are proposed for solving twodimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.
Multigrid Method for 2D Helmholtz Equation using Higher Order Finite Difference Scheme Accelerated by Krylov Subspace
, 2014
"... ABSTRACT A sixthorder compact difference scheme is applied with uniform mesh sizes in different coordinate directions to discretize a two dimensional Helmholtz equation. Multigrid method is designed to solve the resulting sparse linear systems. Numerical results are conducted to test the accuracy ..."
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ABSTRACT A sixthorder compact difference scheme is applied with uniform mesh sizes in different coordinate directions to discretize a two dimensional Helmholtz equation. Multigrid method is designed to solve the resulting sparse linear systems. Numerical results are conducted to test the accuracy and performance of the sixthorder compact difference scheme with multigrid method and to compare it with the standard secondorder difference scheme and fourthorder compact difference scheme. The errors norms L 2 is used to establish efficiency and accuracy of the proposed scheme with multigrid method.
Analysis on two approaches for high order accuracy finite difference computation
, 2012
"... a b s t r a c t We analyze two approaches for enhancing the accuracy of the standard second order finite difference schemes in solving one dimensional elliptic partial differential equations. These are the fourth order compact difference scheme and the fourth order scheme based on the Richardson ex ..."
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a b s t r a c t We analyze two approaches for enhancing the accuracy of the standard second order finite difference schemes in solving one dimensional elliptic partial differential equations. These are the fourth order compact difference scheme and the fourth order scheme based on the Richardson extrapolation techniques. We study the truncation errors of these approaches and comment on their regularity requirements and computational costs. We present numerical experiments to demonstrate the validity of our analysis.
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"... Simple and fast multigrid solution of Poisson's equation using diagonally oriented grids A.J. Roberts (Received 7 Septemeber 2000, revised 13 June 2001) We solve Poisson's equation using new multigrid algorithms that converge rapidly. The feature of the 2D and 3D algorithms are the use of ..."
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Simple and fast multigrid solution of Poisson's equation using diagonally oriented grids A.J. Roberts (Received 7 Septemeber 2000, revised 13 June 2001) We solve Poisson's equation using new multigrid algorithms that converge rapidly. The feature of the 2D and 3D algorithms are the use of diagonally oriented grids in the multigrid hierarchy for a much richer and eective communication between the levels of the multigrid. Numerical investigations into solving Poisson's equation in the unit square and unit cube show simple versions of the proposed algorithms
Bachelorarbeit Efficient generation of Mehrstellenverfahren for elliptic PDEs
"... Quellen angefertigt habe und dass die Arbeit in gleicher oder ähnlicher Form noch keiner anderen Prüfungsbehörde vorgelegen hat und von dieser als Teil einer Prüfungsleistung angenommen wurde. Alle Ausführungen, die wörtlich oder sinngemäß übernommen wurden, sind als solche gekennzeichnet. Der Unive ..."
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Quellen angefertigt habe und dass die Arbeit in gleicher oder ähnlicher Form noch keiner anderen Prüfungsbehörde vorgelegen hat und von dieser als Teil einer Prüfungsleistung angenommen wurde. Alle Ausführungen, die wörtlich oder sinngemäß übernommen wurden, sind als solche gekennzeichnet. Der Universität ErlangenNürnberg, vertreten durch den Lehrstuhl für Systemsimulation (Informatik 10), wird für Zwecke der Forschung und Lehre ein einfaches, kostenloses, zeitlich