ABSTRACT
Chapter 8 deals with various direct and iterative methods for solving the system of equations arising from finite difference/finite volume discretisation. The accuracy of the direct method deteriorates with a large number of equations. Only iterative methods are used for solving large systems of equations. The Gauss-Seidel iterative method converges more quickly than the Gauss-Jordan method. Convergence rate can be further accelerated by using the relaxation parameter. The iterative methods demand diagonal dominance of the system of equations and sometimes symmetry. For a non-orthogonal grid, the diagonal dominance of the coefficient matrix is lost. The use of conjugate gradient (CG) methods improves speed and efficiency. The bi-conjugate and CG square solver are proven to be very robust and become efficient when a pre-conditioner is applied. The convergence behaviour of iterative methods is generally slow, and computational time increases with a higher number of grid points. The multigrid method uses a series of coarse grids to approximate the solution. Lower-frequency waves become higher-frequency waves in coarser grids, which are smoothed out at a faster rate. Also, boundary information travels more quickly to the centre in a coarser grid. Hence, the convergence rate also becomes faster.
