Parallel block preconditioning for the solution of boundary value methods

The discrete problem associated with a two-step boundary value method (BVM) for the solution of initial value problems is a non-symmetric block tridiagonal system. This system may be efficiently solved on a parallel computer by using a conjugate gradient type method with a suitable preconditioning. In this paper we consider a BVM based on an Adams method of order three and the trapezoidal method. The structure of the coefficient matrix allows us to derive good stability properties and an efficient preconditioning. Both the theoretical properties and the parallel implementation are discussed in more detail. In the numerical tests section, the preconditioning has been associated with the Bi-CGSTAB algorithm. The parallel algorithm has been tested on a network of transputers.