A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on nonuniform grids

Ge, Yongbin; Cao, Fujun; Zhang, Jun

doi:10.1016/j.jcp.2012.09.034

Cited by 31 publications

(27 citation statements)

References 38 publications

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“…The first three of these problems are convection diffusion equations and the results of the present method on nonuniform grids are compared with those obtained using the most recent similar work which is available only for uniform grids [4][5][6][7]. For the purpose of comparison with 3D nonuniform grid, we consider the recent work of Ge et al [13] that proposed a non-uniform HOC scheme to solve the 3D Poisson equation. So, our last test problem is a Poisson equation that can be obtained from Eq.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…For the purpose of comparison with existing numerical results, we consider the recent work by Ge et al [13] that introduced HOC scheme for solving the 3D Poisson equation on non-uniform grids. In this problem, we consider the special case of the convection diffusion equation (1) Table 4 gives the maximum absolute errors and the convergence order on uniform and non-uniform grids.…”

Section: Problemmentioning

confidence: 99%

“…Ge and Cao [12] developed a multigrid method based on [11] to solve the 2D convection diffusion equation. Recently, Ge et al [13] proposed a multigrid method for solving the 3D Poisson equation, which is a convectiondiffusion equation with zero convection and constant diffusion coefficients.…”

Section: Introductionmentioning

confidence: 99%

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Non-uniform HOC Scheme for the 3D Convection−Diffusion Equation

Shanab¹

2013

ACM

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In this paper, we extend the work of Kalita et al. [11] to solve the steady 3D convection-diffusion equation with variable coefficients on non-uniform grid. The approach is based on the use of Taylor series expansion, up to the fourth order terms, to approximate the derivatives appearing in the 3D convection diffusion equation. Then the original convection-diffusion equation is used again to replace the resulting higher order derivative terms. This leads to a higher order scheme on a compact stencil (HOC) of nineteen points. Effectiveness of this method is seen from the fact that it can handle the singularity perturbed problems by employing a flexible discretized grid that can be adapted to the singularity in the domain. Four difficult test cases are chosen to demonstrate the accuracy of the present scheme. Numerical results show that the fourth order accuracy is achieved even though the Reynolds number (Re) is high.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Problemmentioning

confidence: 99%

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Non-uniform HOC Scheme for the 3D Convection−Diffusion Equation

Shanab¹

2013

ACM

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“…Under uniform grids distribution the scheme has four to fifth order accuracy as proposed by [25,26,18].…”

Section: Mathematical Formulationmentioning

confidence: 99%

Multigrid solution for the cauchy problem associated with helmholtz type equation on non uniform grids

Ghaffar¹,

Badshah²,

Islam³

2017

Matrix sci. math.

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“…However, the derivation of the high order compact scheme on nonuniform grid is very nontrivial. The high order compact scheme on nonuniform mesh for 2D convection diffusion equation and 3D Poisson equation are presented in [13,14] and [15], respectively. Numerical results show that the HOC scheme on nonuniform grids is very robust and efficient for the convection diffusion problems with boundary layers and local singularities.…”

Section: Introductionmentioning

confidence: 99%

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

Cao¹,

Yuan²,

Ge³

2017

dtcse

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Abstract. High order compact scheme has extensively been used for solving the Helmholtz equation on uniform mesh. Although uniform spacing is computationally advantageous for isotropic problems, the non-uniform mesh will be superior to the problems with the anisotropic properties or boundary layers. In this paper, we proposed a compact scheme on nonuniform mesh which has at least third order accuracy for the 2D Helmholtz equation with variable wave number. A novel adaptive mesh algorithm is also presented to generate the orthogonal nonuniform mesh. The 2D Helmholtz equation is solved by combining the HOC scheme on nonuniform mesh with the moving mesh method. The multigrid method is used to solve the discrete algebraic system. Numerical experiments are executed to verify the accuracy and efficiency of the presented method.

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A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on nonuniform grids

Cited by 31 publications

References 38 publications

Non-uniform HOC Scheme for the 3D Convection−Diffusion Equation

Non-uniform HOC Scheme for the 3D Convection−Diffusion Equation

Multigrid solution for the cauchy problem associated with helmholtz type equation on non uniform grids

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

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