Qualitative properties of modified equations

Abstract

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 modified equation. If the method and the system have a particular qualitative property then it is important to determine whether the modified equations inherit this property. In this article, a technique is introduced for proving that the modified 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-Kutte methods. The new approach unifies and extends results of this type that have been derived by other means: results are presented for integral preservation, reversibility, inheritance of fixed 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 modified equations.