Boundary value problems and dichotomic stability

Abstract

Since the conditioning of a boundary value problem (BVP) is closely related to the existence of a dichotomic fundamental solution (i.e., where one set of modes is increasing and a complementary set is decreasing), it is important to have discretization methods that conserve this dichotomy property. The conditions this imposes on such a method are investigated in this paper. They are worked out in more detail for scalar second-order equations (the central difference scheme), and for linear first-order systems as well; for the latter type both one-step methods (including collocation) and multistep methods (those that may be used in multiple shooting) are examin