Stationary Iterative Methods and Their Application to Some Problems Arising in Physics

The objective of this paper is to evaluate the Stationary Iterative Methods and Their Application to Some Problems Arising in Physics. The solution of such problems depends on the diagonal dominance criterion for the convergence of such stationary iterative methods. The solution of such linear algebraic equations is discussed in this work. The iteration matrices of the Jacobi and Gauss-Seidel methods were derived via the splitting of the coefficient matrices. Their convergence was examined with the diagonal dominance criterion and the schemes were found to converge. The weak diagonal dominance criterion was thereafter derived. The iteration matrices were then used to obtain their respective spectral radii. The rates of convergence of the schemes were discussed in relation to spectral radii of the iteration matrices and structure of the coefficient matrices. The coefficient matrices of the examples used were of three categories: the strictly diagonally dominant, the weakly diagonally dominant and the non-diagonally dominant. The spectral radii of the iteration matrices for the strictly diagonally dominant systems were less than unity and the iterations for Jacobi, Gauss-Seidel and Successive overrelaxation methods converged. For the weakly diagonally dominant systems, only the Jacobi iteration matrix had a spectral radius higher than unity and the scheme did not converge.