Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application

Kedir Aliyi Koroche

doi:doi:10.11648/j.dmath.20210601.12

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Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application

Kedir Aliyi Koroche

Published in International Journal of Discrete Mathematics (Volume 6, Issue 1)

Received: 3 April 2021 Accepted: 6 May 2021 Published: 14 May 2021

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Abstract

In this paper, Peaceman-Rachford alternating direct implicitly methods presented and applied for solve linear advection-diffusion equation. First, the domain was discretized using the uniform mesh of step length and time step. Secondly, by applying the Taylor series methods, we discretize partial derivative of governing equation and we obtain the central difference equation for Partial differential equation of given governing equation in both duration. Then rearranging the obtained central difference equation; we write the two half scheme of the present method. From each half of these schemes, we obtain tri-diagonal coefficient matrices associated with the system of difference equation. Lastly by applying the Thomas algorithm and writing MATLAB code for the scheme we obtain solution of the governing linear advection diffusion equation. To validate the applicability of the proposed method, three model examples are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (L₁-norm) and L₂-norm, numerical error and experimental order of convergence. The stability and convergence of the present numerical method are also guaranteed and the comparability of numerical solution and the stability of the present method are presented by using the graphical and tabular form. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Published in	International Journal of Discrete Mathematics (Volume 6, Issue 1)
DOI	10.11648/j.dmath.20210601.12
Page(s)	5-14
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

Linear Advection-diffusion Equation, Peaceman-Rachford Alternating Direct Implicitly Method, Taylor Series Methods, Tri-diagonal Coefficient Matrices, Thomas Algorithm, Stability and Convergence

References

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[2]	Jiaqi Zhong, Cheng Zeng, Yupeng Yuan, Yuzhe Zhang, Ye Zhang. Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method. AIP Advances, 8 (4), 2018.
[3]	Evstigneev, M. Nikolay. Numerical analysis of Krylov multi-grid methods for stationary advection-diffusion equation. International Journal of Physics: Conference Series, 1391 (1), 2019.
[4]	Soyoon Bak, Philsu Kim, Xiangfan Piao, Sunyoung Bu. Numerical solution of advection-diffusion type equation by modified error corrections scheme. Open Springer journal of Advances in Difference Equations, 2018.
[5]	Yu Liu, Marcial Gonzalez, Carl Wassgren. Modeling Granular Material Blending in a Rotating Drum Using a Finite Element Method and Advection-Diffusion Equation Multi-scale Model. American Institute of Chemical Engineers, 64 (9), 3277-3292, 2018.
[6]	AVCI Derya, Aylin Yetim. Analytical solutions to the advection-diffusion equation with the Atangana-Baleanu derivative over a finite domain", Balıkesir Universities Fen Balmier Enstitüsü Dergisi, 20 (2) 382-395, 2018.
[7]	Abdelkader Mojtabi, Michel Deville. One-dimensional linear advection-diffusion equation: Analytical and finite element solution. Computers and Fluids, Elsevier, 107, 189-195, 2015.
[8]	Cotta RM. Integral transforms in computational heat and fluid flow. Boca Raton (FL): CRC Press; 1993.
[9]	V. A. N. Genuchten, Martinus TH, Feike J. Leij, Todd H. Skaggs, Nobuo Toride, Scott A. Bradford, Elizabeth M. Pontedeiro. Exact analytical solutions for contaminant transport in rivers 1. The equilibrium advection-dispersion equation. Journal of Hydrology and Hydromechanics, 61 (2), 146-160, 2013.
[10]	Guerrero, J. S. Pérez, Luiz Cláudio Gomes Pimentel, Todd H. Skaggs, and M. T. H. Van Genuchten. Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique. International Journal of Heat and Mass Transfer 52 (13) 3297-3304, 2009.
[11]	L.S. Andallah, M. R. Khatun. Numerical solution of advection-diffusion equation using finite difference schemes. Bangladesh Journal of Scientific and Industrial Research, 55 (1) 15-22, 2020.
[12]	W. H. Hundsdorfer, J. G. Verwer. Stability and Convergence of the Peaceman-Rachford alternative direct implicitly method for Initial-Boundary Value Problems. Journal of mathematics of computation, 53 (187), 81-101, 1987.
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[14]	Gurhan Gurarslan, Halil Karahan, Devrim Alkaya, Murat Sari, Mutlu Yasar. Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method. Hindawi Publishing Corporation Mathematical Problems in Engineering. 2013 (7), 2013.
[15]	Sigrun Ortleb. L²-stability analysis of IMEX-(σ,μ) D schemes for linear advection-diffusion equations. Applied Numerical Mathematics, 147, 43–65, 2020.
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Kedir Aliyi Koroche. (2021). Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application. International Journal of Discrete Mathematics, 6(1), 5-14. https://doi.org/10.11648/j.dmath.20210601.12

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ACS Style

Kedir Aliyi Koroche. Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application. Int. J. Discrete Math. 2021, 6(1), 5-14. doi: 10.11648/j.dmath.20210601.12

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AMA Style

Kedir Aliyi Koroche. Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application. Int J Discrete Math. 2021;6(1):5-14. doi: 10.11648/j.dmath.20210601.12

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@article{10.11648/j.dmath.20210601.12,
  author = {Kedir Aliyi Koroche},
  title = {Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application},
  journal = {International Journal of Discrete Mathematics},
  volume = {6},
  number = {1},
  pages = {5-14},
  doi = {10.11648/j.dmath.20210601.12},
  url = {https://doi.org/10.11648/j.dmath.20210601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20210601.12},
  abstract = {In this paper, Peaceman-Rachford alternating direct implicitly methods presented and applied for solve linear advection-diffusion equation. First, the domain was discretized using the uniform mesh of step length and time step. Secondly, by applying the Taylor series methods, we discretize partial derivative of governing equation and we obtain the central difference equation for Partial differential equation of given governing equation in both duration. Then rearranging the obtained central difference equation; we write the two half scheme of the present method. From each half of these schemes, we obtain tri-diagonal coefficient matrices associated with the system of difference equation. Lastly by applying the Thomas algorithm and writing MATLAB code for the scheme we obtain solution of the governing linear advection diffusion equation. To validate the applicability of the proposed method, three model examples are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (L1-norm) and L2-norm, numerical error and experimental order of convergence. The stability and convergence of the present numerical method are also guaranteed and the comparability of numerical solution and the stability of the present method are presented by using the graphical and tabular form. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.},
 year = {2021}
}

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TY  - JOUR
T1  - Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application
AU  - Kedir Aliyi Koroche
Y1  - 2021/05/14
PY  - 2021
N1  - https://doi.org/10.11648/j.dmath.20210601.12
DO  - 10.11648/j.dmath.20210601.12
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 5
EP  - 14
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20210601.12
AB  - In this paper, Peaceman-Rachford alternating direct implicitly methods presented and applied for solve linear advection-diffusion equation. First, the domain was discretized using the uniform mesh of step length and time step. Secondly, by applying the Taylor series methods, we discretize partial derivative of governing equation and we obtain the central difference equation for Partial differential equation of given governing equation in both duration. Then rearranging the obtained central difference equation; we write the two half scheme of the present method. From each half of these schemes, we obtain tri-diagonal coefficient matrices associated with the system of difference equation. Lastly by applying the Thomas algorithm and writing MATLAB code for the scheme we obtain solution of the governing linear advection diffusion equation. To validate the applicability of the proposed method, three model examples are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (L1-norm) and L2-norm, numerical error and experimental order of convergence. The stability and convergence of the present numerical method are also guaranteed and the comparability of numerical solution and the stability of the present method are presented by using the graphical and tabular form. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.
VL  - 6
IS  - 1
ER  -

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Author Information

Kedir Aliyi Koroche

Department of Mathematics, College of Natural and Computational Sciences, Ambo University, Ambo, Ethiopia

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Plain Text BibTeX RIS

APA Style

Kedir Aliyi Koroche. (2021). Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application. International Journal of Discrete Mathematics, 6(1), 5-14. https://doi.org/10.11648/j.dmath.20210601.12

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ACS Style

Kedir Aliyi Koroche. Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application. Int. J. Discrete Math. 2021, 6(1), 5-14. doi: 10.11648/j.dmath.20210601.12

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AMA Style

Kedir Aliyi Koroche. Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application. Int J Discrete Math. 2021;6(1):5-14. doi: 10.11648/j.dmath.20210601.12

Copy | Download

@article{10.11648/j.dmath.20210601.12,
  author = {Kedir Aliyi Koroche},
  title = {Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application},
  journal = {International Journal of Discrete Mathematics},
  volume = {6},
  number = {1},
  pages = {5-14},
  doi = {10.11648/j.dmath.20210601.12},
  url = {https://doi.org/10.11648/j.dmath.20210601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20210601.12},
  abstract = {In this paper, Peaceman-Rachford alternating direct implicitly methods presented and applied for solve linear advection-diffusion equation. First, the domain was discretized using the uniform mesh of step length and time step. Secondly, by applying the Taylor series methods, we discretize partial derivative of governing equation and we obtain the central difference equation for Partial differential equation of given governing equation in both duration. Then rearranging the obtained central difference equation; we write the two half scheme of the present method. From each half of these schemes, we obtain tri-diagonal coefficient matrices associated with the system of difference equation. Lastly by applying the Thomas algorithm and writing MATLAB code for the scheme we obtain solution of the governing linear advection diffusion equation. To validate the applicability of the proposed method, three model examples are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (L1-norm) and L2-norm, numerical error and experimental order of convergence. The stability and convergence of the present numerical method are also guaranteed and the comparability of numerical solution and the stability of the present method are presented by using the graphical and tabular form. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application
AU  - Kedir Aliyi Koroche
Y1  - 2021/05/14
PY  - 2021
N1  - https://doi.org/10.11648/j.dmath.20210601.12
DO  - 10.11648/j.dmath.20210601.12
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 5
EP  - 14
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20210601.12
AB  - In this paper, Peaceman-Rachford alternating direct implicitly methods presented and applied for solve linear advection-diffusion equation. First, the domain was discretized using the uniform mesh of step length and time step. Secondly, by applying the Taylor series methods, we discretize partial derivative of governing equation and we obtain the central difference equation for Partial differential equation of given governing equation in both duration. Then rearranging the obtained central difference equation; we write the two half scheme of the present method. From each half of these schemes, we obtain tri-diagonal coefficient matrices associated with the system of difference equation. Lastly by applying the Thomas algorithm and writing MATLAB code for the scheme we obtain solution of the governing linear advection diffusion equation. To validate the applicability of the proposed method, three model examples are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (L1-norm) and L2-norm, numerical error and experimental order of convergence. The stability and convergence of the present numerical method are also guaranteed and the comparability of numerical solution and the stability of the present method are presented by using the graphical and tabular form. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.
VL  - 6
IS  - 1
ER  -

Copy | Download