Numerical iterative schemes comparative study

Skills or knowledge: Knowledge of various numerical iterative schemes

Software or programs used: MATLAB programming

• Project: The comparison of Jacobi, Gauss Siedel and SOR iterative methods by solving for a pressure Poisson equation with a specified boundary condition in Matlab. The comparison is done by considering the number of iterations taken by the iterative methods to converge.

• Poisson equation:

• d^2 (u) /dx^2 + d^2 (u) /dy^2 = 2(x(x-1) +y(y-1))

• 2(x(x-1) +y(y-1)) = f(x)

• Boundary condition:

• u=0 along all four sides of the square domain considered.

• The convergence iterations required for each method are monitored till they reach the specified error value.

• Jacobi iterative method:

• W=1

• u(m+1)[i,j]= 1/4 (u[i+1,j] + u[i-1,j] + u[i,j-1] + -f(i,j) h^2)

• Gauss Siedel iterative method:

• W=1

• u(m+1)[I,j] = 1/4( u[i+1,j] + u(m+1)[i-1,j] + u[I,j+1] + u(m+1)[I,j-1] –f(I,j) h^2)

• SOR iterative method:

• 1 < W <2

• u(m+1)[I,j] =(1-w) u[I,j] + w/4( u[i+1,j] + u(m+1)[i-1,j] + u[I,j+1] + u(m+1)[I,j-1] –f(I,j) h^2)

• Comparison results:

• The SOR method takes the least number of iterations for convergence to the specified error value. The Jacobi method takes the most number of iterations to converge. The Gauss Siedel iterative method requires a lesser number of iterations than the Jacobi method.