ON QUARTER-SWEEP FINITE DIFFERENCE SCHEME
FOR ONE-DIMENSIONAL POROUS MEDIUM EQUATIONS
Jackel Chew Vui Lung1, Jumat Sulaiman2
1Faculty of Computing and Informatics
Universiti Malaysia Sabah Labuan International Campus
87000, Labuan F.T., MALAYSIA
2Faculty of Science and Natural Resource
Universiti Malaysia Sabah
88400, Kota Kinabalu, Sabah, MALAYSIA

Abstract

In this article, we introduce an implicit finite difference approximation for one-dimensional porous medium equations using Quarter-Sweep approach. We approximate the solutions of the nonlinear porous medium equations by the application of the Newton method and use the Gauss-Seidel iteration. This yields a numerical method that reduces the computational complexity when the spatial grid spaces are reduced. The numerical result shows that the proposed method has a smaller number of iterations, a shorter computation time and a good accuracy compared to Newton-Gauss-Seidel and Half-Sweep Newton-Gauss-Seidel methods.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 3
Year: 2020

DOI: 10.12732/ijam.v33i3.6

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] A. Sunarto, J. Sulaiman, Investigation of fractional diffusion equation via QSGS iterations, J. Phys. Conf. Ser., 1179 (2019), Art. 012014.
  2. [2] A.D. Polyanin and V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equation, Chapman and Hall/CRC Press, Boca Raton (2004).
  3. [3] A.M. Wazwaz, The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations, Comput. Math. Appl., 54 (2007), 933-939.
  4. [4] E. Aruchunan, M.S. Muthuvalu, J. Sulaiman, Quarter-Sweep iteration concept on conjugate gradient normal residual method via second order quadrature - finite difference schemes for solving Fredholm integrodifferential equations, Sains Malaysiana, 44, No 1 (2015), 139-146.
  5. [5] J.H. Eng, A. Saudi, J. Sulaiman, Implementation of Quarter-Sweep approach in poisson image blending problem, In: R. Alfred, Y. Lim, A. Ibrahim, P. Anthony (Eds), Computational Science and Technology, Lecture Notes in Electrical Engineering, Vol. 481, Springer, Singapore (2019).
  6. [6] J.J. More, Global convergence of Newton-Gauss-Seidel methods, SIAM J. Numer. Anal., 8, No 2 (1971), 325-336.
  7. [7] J.L. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, New York (2007).
  8. [8] J.M. Ortega, W.C. Rheinboldt, Monotone iterations for nonlinear equations with application to Gauss-Seidel methods, SIAM J. Numer. Anal., 4, No 2 (1967), 171-190.
  9. [9] J.V.L. Chew, J. Sulaiman, Half-Sweep Newton-Gauss-Seidel for implicit finite difference solution of 1D nonlinear porous medium equations, Global J. Pure Appl. Math., 12, No 3 (2016), 2745-2752.
  10. [10] M. Othman, A.R. Abdullah, An efficient four points modified explicit group Poisson solver, Int. J. Comput. Math., 76 (2000), 203-217.
  11. [11] M.N. Suardi, N.Z.F.M. Radzuan, J. Sulaiman, Performance analysis of Quarter-Sweep Gauss-Seidel iteration with cubic b-spline approach to solve two-point boundary value problems, Adv. Sci. Lett., 24, No 3 (2018), 1732- 1735.
  12. [12] N.M. Nusi, M. Othman, Half- and Quarter-Sweeps implementation of finite-difference time-domain method, Malaysian J. Math. Sci., 3, No 1 (2009), 45-53.