Runge-Kutta discretizations of singularly perturbed gradient equations

Beyn W-J, Schropp J (2000)
BIT Numerical Mathematics 40(3): 415-433.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Beyn, Wolf-JürgenUniBi; Schropp, Johannes
Abstract / Bemerkung
We analyze Runge-Kutta discretizations applied to singularly perturbed gradient systems. It is shown in which sense the discrete dynamics preserve the geometric properties and the longtime behavior of the underlying ordinary differential equation. If the continuous system has an attractive invariant manifold then numerical trajectories started in some neighbourhood (the size of which is independent of the step-size and the stiffness parameter) approach an equilibrium in a nearby manifold. The proof combines invariant manifold techniques developed by Nipp and Stoffer for singularly perturbed systems with some recent results of the second author on the global behavior of discretized gradient systems. The results support the favorable behavior of ODE methods for stiff minimization problems.
Stichworte
singular perturbation problems; Runge-Kutta methods; gradient differential equations; invariant manifolds
Erscheinungsjahr
2000
Zeitschriftentitel
BIT Numerical Mathematics
Band
40
Ausgabe
3
Seite(n)
415-433
ISSN
0006-3835
Page URI
https://pub.uni-bielefeld.de/record/1619077

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Beyn W-J, Schropp J. Runge-Kutta discretizations of singularly perturbed gradient equations. BIT Numerical Mathematics. 2000;40(3):415-433.
Beyn, W. - J., & Schropp, J. (2000). Runge-Kutta discretizations of singularly perturbed gradient equations. BIT Numerical Mathematics, 40(3), 415-433. doi:10.1023/A:1022359511479
Beyn, W. - J., and Schropp, J. (2000). Runge-Kutta discretizations of singularly perturbed gradient equations. BIT Numerical Mathematics 40, 415-433.
Beyn, W.-J., & Schropp, J., 2000. Runge-Kutta discretizations of singularly perturbed gradient equations. BIT Numerical Mathematics, 40(3), p 415-433.
W.-J. Beyn and J. Schropp, “Runge-Kutta discretizations of singularly perturbed gradient equations”, BIT Numerical Mathematics, vol. 40, 2000, pp. 415-433.
Beyn, W.-J., Schropp, J.: Runge-Kutta discretizations of singularly perturbed gradient equations. BIT Numerical Mathematics. 40, 415-433 (2000).
Beyn, Wolf-Jürgen, and Schropp, Johannes. “Runge-Kutta discretizations of singularly perturbed gradient equations”. BIT Numerical Mathematics 40.3 (2000): 415-433.