Arabian Journal of Geosciences

, Volume 6, Issue 9, pp 3409–3415

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

Original Paper

DOI: 10.1007/s12517-012-0591-9

Salmasi, F. & Azamathulla, H.M. Arab J Geosci (2013) 6: 3409. doi:10.1007/s12517-012-0591-9

## Abstract

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.

### Keywords

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

### Notation

ω

Relaxation coefficient

Δx

Computational grid interval in the x direction

Δy

Computational grid interval in the y direction

β

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

K

Hydraulic conductivity

c

Slope of the water table

h

hi,jm

Head in i,j node and time interval m

y0

Assumed initial head at x = 0

s

Maximum length in the x direction

q

Unit width discharge

qx, qy

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

Δt

Time increment

g

Gravitational acceleration

j

jth grid point value

i

ith grid point value