Tech Report CS-02-06

Precisely A(alpha)-stable One-Leg Multistep Methods

Micha Janssen and Pascal Van Hentenryck

April 2002

Abstract:

We consider One-Leg Multistep (OLM) methods for initial value problems in ODEs. These methods are derived from the functional equation p'(t) = f(t,p(t)), where p is a polynomial approximation of the solution. This equation, applied at an evaluation point t_e, produces a nonlinear multistep formula p'(t_e) = f(t_e,p(t_e)) which can be used to compute the solution at the next integration point. We show that there exists a point t* which leads to an OLM formula which is more precise than BDF's, which is (almost) precisely A(alpha)-stable (a concept introduced to capture the ideal stability region) for a k-step method (k <= 6), and whose stability angle is essentially similar to BDF's. We also show how to apply the corrector idea of Klopfenstein to further improve the stability region.

(complete text in pdf or gzipped postscript)