Strang-type preconditioners for systems of LMF-based ODE codes (Q2725334)

This paper is concerned with the solution of the linear algebraic systems which arise in the solution of linear initial value problems NEWLINE\[NEWLINEy'(t) = J y(t) + g(t), \quad y(t_0)= y_0 \in \mathbb{R}^m, \quad t \in [t_0,T],\tag{1}NEWLINE\]NEWLINE by means of the so called boundary value methods introduced by \textit{L. Brugnano} and \textit{D. Trigiante} [Appl. Numer. Math. 13, No. 4, 291-304 (1993; Zbl 0805.65076)]. For a uniform mesh with \(s\) grid points in the interval \( [t_0,T]\) the boundary value method for (1) requires the solution of a sparse linear system of size \( s m \) where the matrix of coefficients has a special block-circulant structure. NEWLINENEWLINENEWLINEIn this context the authors propose Strang type preconditioners tailored for the problems at hand which combined with GMRES method implies the convergence in at most a given number of iterations depending on \(m\) and the linear multistep method that discretizes the differential equation. The paper ends illustrating the effectiveness of the proposed technique for the differential equations which arise in the spatial discretization of the one-dimensional heat equation and the second order wave equation.