Tech Report CS-02-02
Shifted Slope-Comparison Multistep Formulas for ODEs
Micha Janssen and Pascal Van Hentenryck
We present new multistep formulas for initial value problems in ODEs, which we call Shifted Slope-Comparison (SSC) formulas. SSC formulas 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 the choice of t_e is critical for the accuracy and stability of SSC formulas. In particular, we show that there exists a point t+ which leads to an SSC formula with essentially the same accuracy and stability as Adams-Moulton. More interestingly, we show that there exists a point t* which leads to an SSC formula which is more precise than BDF's, which is precisely A(alpha)-stable (a concept which aims at capturing the ideal stability region), and whose stability angle alpha is essentially similar to BDF's. We also show how to apply the corrector idea of Klopfenstein to further improve the stability region.