Tech Report CS-02-06
Precisely A(alpha)-stable One-Leg Multistep Methods
Micha Janssen and Pascal Van Hentenryck
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.