PGSCM: A family of PP-stable Boundary Value Methods for second-order initial value problems

In this paper, we introduce a family of Linear Multistep Methods used as Boundary Value Methods for the numerical solution of initial value problems for second order ordinary differential equations of special type. We rigorously prove that these schemes are PP-stable, in a generalized sense, of arbitrarily high order. This overcomes the barrier that Lambert and Watson established in Lambert and Watson (1976) [1] on Linear Multistep Methods used in the classic way; that is as Initial Value Methods. We call the new methods PGSCMs, an acronym for PνPν-stable Generalized Störmer-Cowell Methods. Numerical illustrations which confirm the theoretical results of the paper are finally given.