On parallel methods for boundary value ODEs

Abstract

Some of the traditional methods for boundary value ODEs, such as standard multiple shooting, finite difference and collocation methods, lend themselves well to parallelization in the independent variable: the first stage of the construction of a solution approximation is performed independently on each subinterval of a mesh. However, the underlying possibly fast bidirectional propagation of information by fundamental modes brings about stability difficulties when information from the different subintervals is combined to form a global solution. Additional difficulties occur when a very stiff problem is to be efficiently and stably solved on a parallel architecture. In this paper difference and parallel shooting methods are examined. A parallel algorithm for the stable solution of the resulting algebraic system is proposed and evaluated. A parallel algorithm for stiff boundary value problems is proposed as well.