ON QUARTER-SWEEP FINITE DIFFERENCE SCHEME
FOR ONE-DIMENSIONAL POROUS MEDIUM EQUATIONS

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

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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.