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Qualitative properties of modified equations

Gonzalez, O. and Higham, D. J. and Stuart, A. M. (1999) Qualitative properties of modified equations. IMA Journal of Numerical Analysis, 19 (2). pp. 169-190. ISSN 0272-4979. doi:10.1093/imanum/19.2.169.

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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.

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Additional Information:© 1999 Institute of Mathematics and its Applications. [Received 7 July 1997 and in revised form 20 April 1998] Work supported by a National Science foundation Graduate Fellowship. Work sup[ported by the Engineering and Physical Sciences Research Council of the UK under grant GR/K80228. Work supported by the National Science Foundation under grant DMS-9201727 and by the Office of Naval Research under grant N00014-92-J-1876.
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NSF Graduate Research FellowshipUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)GR/K80228
Office of Naval Research (ONR)N00014-92-J-1876
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Andrew StuartJ43
Issue or Number:2
Record Number:CaltechAUTHORS:20170609-164025962
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Official Citation:O Gonzalez, DJ Higham, AM Stuart; Qualitative properties of modified equations. IMA J Numer Anal 1999; 19 (2): 169-190. doi: 10.1093/imanum/19.2.169
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78074
Deposited By: George Porter
Deposited On:12 Jun 2017 14:52
Last Modified:15 Nov 2021 17:36

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