A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems

Yongbin Ge; Fei Zhao; Jianying Wei

doi:doi:10.11648/j.acm.20180701.11

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A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems

Yongbin Ge, Fei Zhao, Jianying Wei

Published in Applied and Computational Mathematics (Volume 7, Issue 1)

Received: 3 December 2017 Accepted: 13 December 2017 Published: 12 January 2018

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Abstract

In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa’s HOC ADI method [26], Cao and Ge’s HOC ADI method [27], and our previous exponential HOC ADI method [28].

Published in	Applied and Computational Mathematics (Volume 7, Issue 1)
DOI	10.11648/j.acm.20180701.11
Page(s)	1-10
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), 2018. Published by Science Publishing Group

Keywords

3D Unsteady Convection Diffusion Equation, Rational, High Order Compact Scheme, ADI Method, Stability

References

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APA Style

Yongbin Ge, Fei Zhao, Jianying Wei. (2018). A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems. Applied and Computational Mathematics, 7(1), 1-10. https://doi.org/10.11648/j.acm.20180701.11

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ACS Style

Yongbin Ge; Fei Zhao; Jianying Wei. A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems. Appl. Comput. Math. 2018, 7(1), 1-10. doi: 10.11648/j.acm.20180701.11

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AMA Style

Yongbin Ge, Fei Zhao, Jianying Wei. A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems. Appl Comput Math. 2018;7(1):1-10. doi: 10.11648/j.acm.20180701.11

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@article{10.11648/j.acm.20180701.11,
  author = {Yongbin Ge and Fei Zhao and Jianying Wei},
  title = {A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {1},
  pages = {1-10},
  doi = {10.11648/j.acm.20180701.11},
  url = {https://doi.org/10.11648/j.acm.20180701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180701.11},
  abstract = {In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa’s HOC ADI method [26], Cao and Ge’s HOC ADI method [27], and our previous exponential HOC ADI method [28].},
 year = {2018}
}

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TY  - JOUR
T1  - A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems
AU  - Yongbin Ge
AU  - Fei Zhao
AU  - Jianying Wei
Y1  - 2018/01/12
PY  - 2018
N1  - https://doi.org/10.11648/j.acm.20180701.11
DO  - 10.11648/j.acm.20180701.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 1
EP  - 10
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20180701.11
AB  - In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa’s HOC ADI method [26], Cao and Ge’s HOC ADI method [27], and our previous exponential HOC ADI method [28].
VL  - 7
IS  - 1
ER  -

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Author Information

Yongbin Ge

Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, China
Fei Zhao

Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, China
Jianying Wei

Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, China

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APA Style

Yongbin Ge, Fei Zhao, Jianying Wei. (2018). A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems. Applied and Computational Mathematics, 7(1), 1-10. https://doi.org/10.11648/j.acm.20180701.11

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ACS Style

Yongbin Ge; Fei Zhao; Jianying Wei. A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems. Appl. Comput. Math. 2018, 7(1), 1-10. doi: 10.11648/j.acm.20180701.11

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AMA Style

Yongbin Ge, Fei Zhao, Jianying Wei. A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems. Appl Comput Math. 2018;7(1):1-10. doi: 10.11648/j.acm.20180701.11

Copy | Download

@article{10.11648/j.acm.20180701.11,
  author = {Yongbin Ge and Fei Zhao and Jianying Wei},
  title = {A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {1},
  pages = {1-10},
  doi = {10.11648/j.acm.20180701.11},
  url = {https://doi.org/10.11648/j.acm.20180701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180701.11},
  abstract = {In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa’s HOC ADI method [26], Cao and Ge’s HOC ADI method [27], and our previous exponential HOC ADI method [28].},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems
AU  - Yongbin Ge
AU  - Fei Zhao
AU  - Jianying Wei
Y1  - 2018/01/12
PY  - 2018
N1  - https://doi.org/10.11648/j.acm.20180701.11
DO  - 10.11648/j.acm.20180701.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 1
EP  - 10
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20180701.11
AB  - In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa’s HOC ADI method [26], Cao and Ge’s HOC ADI method [27], and our previous exponential HOC ADI method [28].
VL  - 7
IS  - 1
ER  -

Copy | Download