Qualitative properties of modi"ed equations

Suppose that a consistent one-step numerical method of order r is applied to a smooth system of ordinary differential equations. Given any integer m > 1, the method may be shown to be of order r + m as an approximation to a certain modi"ed equation. If the method and the system have a particular qualitative property then it is important to determine whether the modi"ed equations inherit this property. In this article, a technique is introduced for proving that the modi"ed equations inherit qualitative properties from the method and the underlying system. The technique uses a straightforward contradiction argument applicable to arbitrary one-step methods and does not rely on the detailed structure of associated power series expansions. Hence the conclusions apply, but are not restricted, to the case of Runge–Kutta methods. The new approach uni"es and extends results of this type that have been derived by other means: results are presented for integral preservation, reversibility, inheritance of "xed points, Hamiltonian problems and volume preservation. The technique also applies when the system has an integral that the method preserves not exactly, but to order greater than r . Finally, a negative result is obtained by considering a gradient system and gradient numerical method possessing a global property that is not shared by the associated modi"ed equations.