Runge-Kutta Discretizations of Singularly Perturbed Gradient Equations
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| Type of Publication: | Journal article |
| Publication status: | Published |
| Author: | Beyn, Wolf-Jürgen; Schropp, Johannes |
| Year of publication: | 2000 |
| Published in: | BIT - Numerical Mathematics ; 40 (2000), 3. - pp. 415-433. - ISSN 0006-3835. - eISSN 1572-9125 |
| DOI (citable link): | https://dx.doi.org/10.1023/A:1022359511479 |
| Summary: |
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.
|
| Subject (DDC): | 510 Mathematics |
| Keywords: | Gradient differential equations; singular perturbation problems; invariant manifolds; Runge-Kutta methods |
| Bibliography of Konstanz: | Yes |
| Refereed: | Unknown |
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BEYN, Wolf-Jürgen, Johannes SCHROPP, 2000. Runge-Kutta Discretizations of Singularly Perturbed Gradient Equations. In: BIT - Numerical Mathematics. 40(3), pp. 415-433. ISSN 0006-3835. eISSN 1572-9125. Available under: doi: 10.1023/A:1022359511479
@article{Beyn2000Runge-43181, title={Runge-Kutta Discretizations of Singularly Perturbed Gradient Equations}, year={2000}, doi={10.1023/A:1022359511479}, number={3}, volume={40}, issn={0006-3835}, journal={BIT - Numerical Mathematics}, pages={415--433}, author={Beyn, Wolf-Jürgen and Schropp, Johannes} }
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