| Realistic simulations of convection often require the use of discretized 3-D models, which give rise to systems of linear equations, Ax = b, whose solution calls for the use of reliable, efficient, and robust iterative techniques implemented on fast parallel computers. As convection becomes the dominant transport mechanism in these simulations, however, the coefficient matrix A, which, in addition to the entries due to convection, may also contain entries due to diffusion and reaction, becomes progressively more non-symmetric, non-diagonally-dominant, and indefinite. These matrix properties approach the threshold at which the performance of methods commonly used to solve linear equations either fails or deteriorates.; Consequently, a geometric method for solving linear equations has been developed that possesses the characteristics necessary for an iterative solver to be reliable, efficient, and robust in solving the linear equations arising in simulations of convection, namely, (1) the performance of the method is insensitive to the properties of the spectrum of A, and thus, the method requires no preconditioning; (2) the method is insensitive to the loss of symmetry and loss of diagonal dominance in A; and, (3) the method is insensitive to the degree of non-diagonal dominance of A.; The developed iterative method of oblique projections minimizes the residual vector, r = b − Ax, associated with the linear equations by means of oblique projections of the residual vector, r, onto the planes determined by the column vectors of A, followed by a product of dilative reflections of the projections about a line through the origin of the system of coordinates. It is shown that the shape of the solution trajectories generated by the oblique projections of r depends only on the structure of the column vectors of A, and that, when the column space consists of independent planes in the Euclidean space, the method converges hyperlinearly to machine accuracy after two or three cycles through the column vectors followed by a set of one-dimensional secant extrapolation steps. |
