The Hodie Method for Ordinary Differential Equations

This paper describes a method of generating high order difference approxith matians to an m — order differential operator L using (m + 1) pointsIt uses certain auxiliary points for each (m + 1)-tuple in order to achieve an arbitrary specified order of truncation error in the difference approximation. In the context of solving Lu = f, this method may be interpreted as using L^u = I^f where is an (m + l)-step difference operator and is an expansion of the identity. The acronym HODIE (High Order Differences via Identity Expansion) comes from this interpretation. Under natural assumptions we prove the existance of such approximations, and establish the order of their truncation error and discretization error. We also show the existance of certain Gauss-type points where even higher orders of convergence are achieved. We give an operations count comparison of this new method with five others which shows the HODIE method to be among the most efficient. A brief selection from extensive experiments is given which supports the practicality of the method.