Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of order r is applied t o a smooth system of ordinary differential equations. Given any integer m g reater than or equal to 1, the method may be shown to be of order r + m as an approximation to a certain modified equation. If me method and the syste m have a particular qualitative property then it is important to determine whether the modified equations inherit this property. In this article, a te chnique is introduced for proving that the modified equation's inherit qual itative 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 powe r series expansions. Hence the conclusions apply, but are not restricted, t o the case of Runge-Kutta methods. The new approach unifies and extends res ults of this type that have been derived by other means: results are presen ted for integral preservation, reversibility, inheritance of fixed points, Hamiltonian problems and volume preservation. The technique also applies wh en the system has an integral that the method preserves not exactly, but to order greater than r. Finally, a negative result is obtained by considerin g a gradient system and gradient numerical method possessing a global prope rty that is not shared by the associated modified equations.