Numerical treatment of O.D.Es.: The theory ofA-methods

Abstract

Almost all commonly used methods for O.D.Es. and their most miscellaneous compositions areA-methods, i.e. they can be reduced toz o=?;z j =Az j?1 +h?(x j?1 ,z j?1 ,z j ;h),z j ?? s ,A??(s,s),j=1(1)m. This paper presents a general theory forA-methods and discusses its practical consequences. An analysis of local discretization error (l.d.e.) accumulation results in a general order criterium and reveals which part of the l.d.e. effectively influences the global error. This facilitates the comparison of methods and generalizes considerably the concept of error constants. It is shown, as a consequence, that the global error cannot be safely controlled by the size of the l.d.e. and that the conventional error control may fail in important cases. Furthermore, Butcher's "effective order" methods, the concept of Nordsieck forms, and Gear's interpretation of lineark-step schemes as relaxation methods are generalized. The stability of step changing is shortly discussed.