Iterative parallel methods for boundary value problems

Abstract

A bordered almost block diagonal system (BABD) results from discretizing and linearizing ordinary differential equation (ODE) boundary value problems (BVPs) with nonseparated boundary conditions (BCs) by either spline collocation, finite differences, or multiple shooting. After interval condensation, if necessary, this BABD system reduces to a standard finite difference BABD structure. This system can be solved either using a divide-and-conquer approach or an iterative scheme such as preconditioned conjugate gradients (PCG). Preconditioners approximating the inverse of the finite difference operator are effective and can be computed and applied efficiently in a parallel environment. We present theoretical computational costs, comparing direct and iterative methods, and numerical results computed on a Sequent Symmetry shared memory computer. These demonstrate that the PCG method can outperform the divide-and-conquer approach on systems with many processors when approximately large differential systems. Also, the PCG method scales up better than the implemented divide-and-conquer method. >