Picture of person typing on laptop with programming code visible on the laptop screen

World class computing and information science research at Strathclyde...

The Strathprints institutional repository is a digital archive of University of Strathclyde's Open Access research outputs. Strathprints provides access to thousands of Open Access research papers by University of Strathclyde researchers, including by researchers from the Department of Computer & Information Sciences involved in mathematically structured programming, similarity and metric search, computer security, software systems, combinatronics and digital health.

The Department also includes the iSchool Research Group, which performs leading research into socio-technical phenomena and topics such as information retrieval and information seeking behaviour.

Explore

Qualitative properties of modified equations

Gonzalez, O. and Stuart, A.M. and Higham, D.J. (1999) Qualitative properties of modified equations. IMA Journal of Numerical Analysis, 19 (2). pp. 169-190. ISSN 0272-4979

Full text not available in this repository. Request a copy from the Strathclyde author

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.