An Approximate Minimal Elimination Ordering Scheme

Matrix ordering is a key technique when applying Cholesky factorization method to solving sparse symmetric positive definite system Ax = b. In view of some known minimal elimination ordering methods, an efficient heuristic approximate minimal elimination ordering scheme is proposed, which has the total running time of O(n+m). It is noteworthy that the algorithm can not only find a good ordering efficiently, but also achieve the result of symbolic factorization simultaneously.

A parallel Cholesky algorithm for the solution of symmetric linear systems

For the solution of symmetric linear systems, the classical Cholesky method has proved to be difficult to parallelize. In the present paper, we first describe an elimination variant of Cholesky method to produce a lower triangular matrix which reduces the coefficient matrix of the system to an identity matrix. Then, this elimination method is combined with the partitioning method to obtain a parallel Cholesky algorithm. The total serial arithmetical operations count for the parallel algorithm is of the same order as that for the serial Cholesky method. The present parallel algorithm could thus perform withefficiencyclose to 1 if implemented on a multiprocessor machine. We also discuss theexistenceof the parallel algorithm; it is shown that for a symmetric and positive definite system, the presented parallel Cholesky algorithm is well defined and will run to completion.

Variational analysis for simulating free-surface flows in a porous medium

A variational formulation has been developed to solve a parabolic partial differential equation describing free-surface flows in a porous medium. The variational finite element method is used to obtain a discrete form of equations for a two-dimensional domain. The matrix characteristics and the stability criteria have been investigated to develop a stable numerical algorithm for solving the governing equation. A computer programme has been written to solve a symmetric positive definite system obtained from the variational finite element analysis. The system of equations is solved using the conjugate gradient method. The solution generates time-varying hydraulic heads in the subsurface. The interfacing free surface between the unsaturated and saturated zones in the variably saturated domain is located, based on the computed hydraulic heads. Example problems are investigated. The finite element solutions are compared with the exact solutions for the example problems. The numerical characteristics of the finite element solution method are also investigated using the example problems.