A parallel preconditioning technique for boundary value methods (Q1315826)

This paper considers the boundary value method described in the authors' previous paper [ibid. 13, No. 4, 291-304 (1993; Zbl 0805.65076)]. For the constant coefficient initial value problem \(y'=Ly+b(t)\), \(y(t_ 0)=y_ 0\) a boundary value method approximates the solution by solving a linear system arising from the repeated application of a two-step difference scheme followed by a single application of a one-step difference scheme. As for the previous paper, a sequence of geometrically increasing stepsizes is considered. This paper is concerned with efficient solution of the linear system by preconditioning the matrix. Selection of the preconditioner is motivated by approximating the differential operator leading to the linear system. Parallel implementation and matrix transformation for efficiency are discussed. Experience with three large scale test problems and comparison with the solutions obtained using LSODE are indicated. Since the convergence of the approximate solution to the continuous solution is in some doubt for the stepsize sequence considered (see previous review), the utility of this technique may only be realized after suitably modifying the stepsize sequence to ensure convergence.