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

Rabab Ahmed Shanab; Laila Fouad Seddek; Salwa Amin Mohamed

doi:doi:10.11648/j.acm.20130203.11

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

Rabab Ahmed Shanab, Laila Fouad Seddek, Salwa Amin Mohamed

Received: 13 May 2013 Accepted: Published: 30 June 2013

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Abstract

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.

DOI	10.11648/j.acm.20130203.11
Published in	Applied and Computational Mathematics (Volume 2, Issue 3, June 2013)
Page(s)	64-77
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

3D Convection–Diffusion Equation; Variable Coefficient; Fourth Order Compact Scheme; Non-Uniform Grids; Algebraic Multigrid

References

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[3]	Y. Wang, J. Zhang, Integrated fast and high accuracy computation of convection diffusion equations using multiscale multigrid method, Numerical Methods for Partial Differential Equations, 27(2):399-414, 2011
[4]	L. Ge, J. Zhang, Symbolic computation of high order compact difference for three dimensional linear elliptic partial differential equations with variable schemes coefficients, Journal of Computational and Applied Mathematics 143 (2002) 9–27
[5]	M. gupta, J. zhang, High accuracy multigrid solution of the 3D Convection Diffusion equation, Applied Mathematics and Computation 113 (2000) 249–274
[6]	Y. Wang, J. Zhang, Fast and robust sixth-order multigrid computation for the three-dimensional Convection-Diffusion equation, Journal of Computational and Applied Mathematics 234, pp.3496-3506, 2010.
[7]	J. Zhang, An explicit fourth-order compact finite difference scheme for three-dimensional Convection–Diffusion equation, Communications in Numerical Methods in Engineering, Vol. 14, 263--280 (1998).
[8]	Y. Ge, Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D poisson equation, Journal of Computational Physics 229 (2010) 6381–6391
[9]	Y. Ma, Y. Ge, A high order finite difference method with Richardson extrapolation for 3D Convection Diffusion equation, Applied Mathematics and Computation 215 (2010) 3408–3417
[10]	L. Ge, J, Zhang, High Accuracy Iterative Solution of Convection Diffusion Equation with Boundary Layers on Non-uniform Grids, Journal of Computational Physics171,560–578 (2001)
[11]	J. Kalita, A. Dass, D. Dalal, A transformation-free HOC scheme for steady Convection–Diffusion on non-uniform grids, Int. J. Numer. Meth. Fluids 44 (2004) 33–53
[12]	Y. Ge, F. Cao, Multigrid method based on the transformation-free HOC scheme on non-uniform grids for 2D Convection Diffusion problems, Journal of Computational Physics 230 (2011) 4051–4070
[13]	Y. Ge, F. Cao, J. Zhang, A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on non-uniform grids, Journal of Computational Physics (2012)
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[15]	W. Hachbusch, V. Trottenberg, Multigrid Methods Lecture Notes in Mathematics, Vol. 960, Springer: Berlin, 1982.
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[19]	C. Tien Wu and H. C. Elman, Analysis and Comparison of Geometric and Algebraic Multigrid for Convection‐Diffusion Equations, SIAM J. Sci. Comput., Vol.28, pp.2208–2228, 2006.
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Rabab Ahmed Shanab, Laila Fouad Seddek, Salwa Amin Mohamed. (2013). Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation. Applied and Computational Mathematics, 2(3), 64-77. https://doi.org/10.11648/j.acm.20130203.11

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ACS Style

Rabab Ahmed Shanab; Laila Fouad Seddek; Salwa Amin Mohamed. Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation. Appl. Comput. Math. 2013, 2(3), 64-77. doi: 10.11648/j.acm.20130203.11

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AMA Style

Rabab Ahmed Shanab, Laila Fouad Seddek, Salwa Amin Mohamed. Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation. Appl Comput Math. 2013;2(3):64-77. doi: 10.11648/j.acm.20130203.11

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@article{10.11648/j.acm.20130203.11,
  author = {Rabab Ahmed Shanab and Laila Fouad Seddek and Salwa Amin Mohamed},
  title = {Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation},
  journal = {Applied and Computational Mathematics},
  volume = {2},
  number = {3},
  pages = {64-77},
  doi = {10.11648/j.acm.20130203.11},
  url = {https://doi.org/10.11648/j.acm.20130203.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20130203.11},
  abstract = {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.},
 year = {2013}
}

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TY  - JOUR
T1  - Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation
AU  - Rabab Ahmed Shanab
AU  - Laila Fouad Seddek
AU  - Salwa Amin Mohamed
Y1  - 2013/06/30
PY  - 2013
N1  - https://doi.org/10.11648/j.acm.20130203.11
DO  - 10.11648/j.acm.20130203.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 64
EP  - 77
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20130203.11
AB  - 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.
VL  - 2
IS  - 3
ER  -

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Author Information

Rabab Ahmed Shanab

Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt
Laila Fouad Seddek

Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt
Salwa Amin Mohamed

Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt

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APA Style

Rabab Ahmed Shanab, Laila Fouad Seddek, Salwa Amin Mohamed. (2013). Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation. Applied and Computational Mathematics, 2(3), 64-77. https://doi.org/10.11648/j.acm.20130203.11

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ACS Style

Rabab Ahmed Shanab; Laila Fouad Seddek; Salwa Amin Mohamed. Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation. Appl. Comput. Math. 2013, 2(3), 64-77. doi: 10.11648/j.acm.20130203.11

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AMA Style

Rabab Ahmed Shanab, Laila Fouad Seddek, Salwa Amin Mohamed. Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation. Appl Comput Math. 2013;2(3):64-77. doi: 10.11648/j.acm.20130203.11

Copy | Download

@article{10.11648/j.acm.20130203.11,
  author = {Rabab Ahmed Shanab and Laila Fouad Seddek and Salwa Amin Mohamed},
  title = {Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation},
  journal = {Applied and Computational Mathematics},
  volume = {2},
  number = {3},
  pages = {64-77},
  doi = {10.11648/j.acm.20130203.11},
  url = {https://doi.org/10.11648/j.acm.20130203.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20130203.11},
  abstract = {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.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation
AU  - Rabab Ahmed Shanab
AU  - Laila Fouad Seddek
AU  - Salwa Amin Mohamed
Y1  - 2013/06/30
PY  - 2013
N1  - https://doi.org/10.11648/j.acm.20130203.11
DO  - 10.11648/j.acm.20130203.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 64
EP  - 77
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20130203.11
AB  - 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.
VL  - 2
IS  - 3
ER  -

Copy | Download