On the 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 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.

Backward Error Analysis for Numerical Integrators

“…, ∞, possess F as a first integral. The same result was recently derived by Gonzalez, Higham, and Stuart [13] using a contradiction argument. Example 2.…”

“…In section 3, it will be shown that this leads to a simple characterization of the geometric properties of the modified vector fields and, in section 4, explicit estimates for the difference between the time-one-flow map of the modified vector field X i (δt) and the numerical method will be given. We like to point out that these results can also be (and have been [16,5,18,20,13]) derived using an explicit Taylor series expansion of the flow map and the numerical method. However, we feel that the application of the recursion (2.4)-(2.5) leads to a simplification in the presentation of these results.…”

“…Besides symplectic methods, special instances of these geometric aspects have been discussed before. See, for example, the papers by Reich [26]; Hairer and Stoffer [20]; and Gonzalez, Higham, and Stuart [13].…”

Efficient Numerical Shadowing Global Error Estimation for High Dimensional Dissipative Systems

“…Backward error analysis asks that the computed solution be the solution to a nearby problem and considers a slightly perturbed problem with the same initial conditions. This idea is related to the concept of modified equations [19]. The third type of error analysis, the type with which we concern ourselves, is shadowing.…”

Rigorous Shadowing of Numerical Solutions of Ordinary Differential Equations by Containment

“…Thus, although the orbit of a planet may be reproduced correctly by the numerical trajectory, the time at which a real and simulated planet pass through a fixed plane perpendicular to the orbit may differ. This is the case even if the integrator is symplectic Gonzalez and Stuart 1996). Thus, when attempting to shadow a numerical solution of an ODE, it may be necessary to "rescale" time (Coomes, Koçak, and Palmer 1994b, 1995a, 1995bVan Vleck 1995).…”