Successive Over Relaxation (SOR) of Finite Difference Method solution to Laplace's Equation

Authors - Sathya Swaroop Ganta, Kayatri, Pankaj Arora, Sumanthra Chaudhuri, Projesh Basu, Nikhil Kumar CS

Course - Computational Electromegnetics

Instructor - Dr. Ananth Krishnan
Assistant Professor
Department of Electrical Engineering
Indian Institute of Technology Madras
Chennai, India

Description:-
Objective of the program is to solve for the steady state voltage distribution in a region 0<x<30, 0<y<30, given that one of the sides of square is excited with a voltage of 45*(x/xmax)*((1-x)/xmax)* Volts (xmax=30) and all other sides are maintained at 0 Volts. This voltage at the boundary is symmetrical with its maximum value at centre of the boundary namely x=15. At any iteration, the value of voltage is updated as average of voltages of 4 nearest neighbors, until between consecutive iterations, the error is less than 0.001 V.

In the Successive Over-Relaxation (SOR) technique the matrix update after each iteration is done in a different way. In the SOR method, the current matrix summed with alpha times the difference between the two matrices is updated as the current matrix. The process is repeated from alpha=1 to alpha=1.9.
The imagesc movie for the converging solution is shown for alpha=1.8. The iterations are counted for each alpha and it is observed that the no of iterations are least for alpha=1.8.