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Runge–Kutta Methods for Dissipative and Gradient Dynamical Systems

Humphries, A. R. and Stuart, A. M. (1994) Runge–Kutta Methods for Dissipative and Gradient Dynamical Systems. SIAM Journal on Numerical Analysis, 31 (5). pp. 1452-1485. ISSN 0036-1429.

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The numerical approximation of dissipative initial value problems by fixed time-stepping Runge–Kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared. A general class of ordinary differential equations, for which dissipativity is induced through an inner product, is studied throughout. This class arises naturally in many finite dimensional applications (such as the Lorenz equations) and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics. It is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step-size h > 0. The numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor A_h exists; upper-semicontinuity of A_h at h = 0 is established, which shows that, for h small, every point on the numerical attractor is close to a point on the true global attractor A. Under the additional assumption that the problem is globally Lipschitz, it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of A_h at h = 0 is again established. For gradient systems with globally Lipschitz vector fields it is shown that any Runge–Kutta method preserves the gradient structure for h sufficiently small. For general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small. Convergence of the numerical attractor is studied and, for a dissipative gradient system with hyperbolic equilibria, lower semicontinuity at h = 0 is established. Thus, for such a system, A_h converges to A in the Hausdorff metric as h → 0.

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Additional Information:© 1994 Society for Industrial and Applied Mathematics. Submitted: 16 November 1992. Accepted: 23 August 1993. The authors would like to thank J.F.B.M. Kraaijevanger for useful discussions.
Subject Keywords:Runge-Kutta methods, dynamical systems, dissipativity, gradient systems, attractors
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ26
Issue or Number:5
Classification Code:AMS subject classifications. 34C35, 34D05, 65L07, 65L20
Record Number:CaltechAUTHORS:20170613-084043889
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Official Citation:Runge–Kutta Methods for Dissipative and Gradient Dynamical Systems A. R. Humphries and A. M. Stuart SIAM Journal on Numerical Analysis 1994 31:5, 1452-1485
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78150
Deposited By: Ruth Sustaita
Deposited On:13 Jun 2017 16:23
Last Modified:03 Oct 2019 18:05

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