Arabian Journal of Geosciences

, Volume 6, Issue 9, pp 3409–3415 | Cite as

Determination of optimum relaxation coefficient using finite difference method for groundwater flow

  • Farzin Salmasi
  • Hazi Mohammad Azamathulla
Original Paper


Solution of Laplace’s equation can be done by iteration methods likes Jacobi, Gauss–Seidel, and successive over-relaxation (SOR). There is no new knowledge about the relaxation coefficient (ω) in SOR method. In this paper, we used SOR for solving Laplace’s differential equation with emphasis to obtaining the optimum (minimum) number of iterations with variations of the relaxation coefficient (ω). For this purpose, a code in FORTRAN language has been written to show the solution of a set of equations and its number of iterations. The results demonstrate that the optimum value of ω with minimum iterations is achieved between 1.7 and 1.9. Also, with increasing β = ∆x/∆y from 0.25 to 10, the number of iterations reduced and the optimum value is obtained for β = 2.


Laplace equation Numerical Successive over-relaxation (SOR) Explicit Finite difference Groundwater 



Relaxation coefficient


Computational grid interval in the x direction


Computational grid interval in the y direction


\( \Delta x/\Delta y\,{\text ratio} \,of\,\Delta x\,{\text{per}}\,\Delta y \)


Hydraulic conductivity


Slope of the water table


Groundwater head


Head in i,j node and time interval m


Assumed initial head at x = 0


Maximum length in the x direction


Unit width discharge

qx, qy

Discharges per unit width in the x and y directions, respectively


Time increment


Gravitational acceleration


jth grid point value


ith grid point value


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Copyright information

© Saudi Society for Geosciences 2012

Authors and Affiliations

  1. 1.Department of Water Engineering, Faculty of AgricultureUniversity of TabrizTabrizIran
  2. 2.River Engineering and Urban Drainage (REDAC)Universiti Sains MalaysiaSeri AmpanganMalaysia

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