A COMPARISON OF SERIAL AND PARALLEL SOLUTIONS OF TWO DIMENSIONAL HEAT CONDUCTION EQUATION

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Published Dec 29, 2020
DOI: https://doi.org/10.53874/jmar.v5i0.79

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S. Abbas
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing China
A. A. Khan
Diamond jubilee Institute, Hyderabad Hunza
B. Shakia
Elysian Academy, Gahkuch Ghizer

Abstract

We study a comparison of serial and parallel solution of 2D-parabolic heat conduction equation using a Crank-Nicolson method with an Alternating Direction Implicit (ADI) scheme. The two-dimensional Heat equation is applied on a thin rectangular aluminum sheet. The forward difference formula is used for time and an averaged second order central difference formula for the derivatives in space to develop the Crank-Nicolson method. FORTRAN serial codes and parallel algorithms using OpenMP are used. Thomas tridigonal algorithm and parallel cyclic reduction methods are employed to solve the tridigonal matrix generated while solving heat equation. This paper emphasize on the run time of both algorithms and their difference. The results are compared and evaluated by creating GNU-plots (Command-line driven graphing utility).

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ABBAS, S.; KHAN, A. A.; SHAKIA, B.. A COMPARISON OF SERIAL AND PARALLEL SOLUTIONS OF TWO DIMENSIONAL HEAT CONDUCTION EQUATION. Journal of Mountain Area Research, [S.l.], v. 5, p. 36-42, dec. 2020. ISSN 2518-850X. Available at: <https://journal.kiu.edu.pk/index.php/JMAR/article/view/79>. Date accessed: 25 apr. 2024. doi: https://doi.org/10.53874/jmar.v5i0.79.
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Issue
Vol 5 (2020)
Section
Mathematical Sciences
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References

[1] T.N. Narasimhan, Fourier heat conduction: History, influence and connections. In: Rev. Geophys, pp. 151-172, (1999).
[2] J. Crank P. Nicolson, A practical method for numerical evaluation of solutions of heat conduction type. In: Cambridge Philosophical Society, pp. 50-64, (1947).
[3] D. Peaceman, M. Rachford, The numerical solution of parabolic and elliptic differential equations, In: J. SIAM 3, pp. 28-41, (1955).
[4] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in FORTRAN 90: The Art of Parallel Scientific Computing. The Pitt Building, Trumpington Street, Cambridge: Cambridge University Press, (1996).
[5] H. A. Luther, B. Carnahan and J. O. Wilkes, Applied Numerical Methods, in: John Wiley and Sons, Inc, pp. 440-446, (1969).
[6] R.W. Hockney. A fast direct solution of Poisson's equation using Fourier analysis, In: Journal of Association or Computing Machinery 12, pp. 95-113, (1965).
[7] R.W. Hockney C.R. Jesshope, Parallel Computers, In: Adam Hilger Ltd (1981).
[8] S. C. Chapra and R. P. Canale, Numerical Methods for Engineers. 2 Penn Plaza, New York, NY 10121: McGraw-Hill Education, pp. 853-885, (2015).
[9] C.R Jesshope, R .W Hockney, Parallel Computers 2. 6000 Broken Sound Parkway
NW, Suite 300 Boca Raton, FL 33487-2742: CRC, Press Taylor Francis Group, pp. 1-6, (2019).
[10] M. Santiago, D. R. Kincaid, Using Cyclic Reduction on a parallel computer to improve the performance of an underwater sound implicit finite difference model, In: Computers Math. Applic. Vol. 21, No. 5, pp. 83-94, (1991).