nach oben

BIT Numerical Mathematics

Erschienen in:

01.12.2015

One-sided direct event location techniques in the numerical solution of discontinuous differential systems

verfasst von: Luca Dieci, Luciano Lopez

Erschienen in: BIT Numerical Mathematics | Ausgabe 4/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper, event location techniques for a differential system the solution of which is directed towards a manifold \(\varSigma \) defined as the 0-set of a smooth function \(h: \varSigma =\{x\in \mathbb {R}^n\,:\, h(x)=0 \}\) are considered. It is assumed that the exact solution trajectory hits \(\varSigma \) non-tangentially, and numerical techniques guaranteeing that the trajectory approaches \(\varSigma \) from one side only (i.e., does not cross it) are studied. Methods based on Runge Kutta schemes which arrive to \(\varSigma \) in a finite number of steps are proposed. The main motivation of this paper comes from integration of discontinuous differential systems of Filippov type, where location of events is of paramount importance.

Vorheriger Artikel Local discontinuous Galerkin methods for fractional ordinary differential equations

Nächster Artikel On the relation between the randomized extended Kaczmarz algorithm and coordinate descent

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

The word direct refers to locating an event point in a finite number of steps, as opposed to an iteration required for this task.

Note that this result does not guarantee that \(h(x_k)\) won’t become positive for some \(k\).

Of course, we are assuming that (1.4) holds.

Acary, V., Brogliato. B.: Numerical methods for monsmooth dynamical systems. In: Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics. Springer, Berlin (2008)

Acary, V., Brogliato, B., Orlov, Y.: Chattering-free digital sliding-mode control with state observer and disturbance rejection. IEEE Trans. Automat. Contr. 57(5), 1087–1101 (2012)

Berardi, M., Lopez, L.: On the continuous extension of Adams–Bashforth methods and the event location in discontinuous ODEs. Appl. Math. Lett. 25(6), 995–999 (2012)CrossRefMathSciNetMATH

Boiko, I., Fridman, L.: Analysis of chattering in continuous sliding-mode controllers. IEEE Trans. Autom. Control 50, 1442–1446 (2005)CrossRefMathSciNet

Calvo, M., Montijano, J.L., Randez, L.: On the solution of discontinuous IVPs by adaptive Runge-Kutta codes. Numer. Algorithms 33, 163–182 (2003)CrossRefMathSciNetMATH

Chau, K.T., Wei, X.X., Gou, X.: Experimental and theoretical simulations of seismic poudings between two adiacient structures. Earthq. Eng. Struct. Dyn. 32, 537–554 (2003)CrossRef

Cooper, G.J.: Stability of Runge-Kutta methods for trajectory problems IMA. J. Numer. Anal. 7, 1–13 (1987)CrossRefMATHMathSciNet

Dieci, L., Lopez, L.: Sliding motion in Filippov differential systems: theoretical results and a computational approach. SIAM J. Numer. Anal. 47, 2023–2051 (2009)CrossRefMathSciNetMATH

Dieci, L., Lopez, L.: A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side. J. Comput. Appl. Math. 236, 3967–3991 (2012)CrossRefMathSciNetMATH

10.

Dieci, L., Lopez, L.: Numerical solution of discontinuous differential systems: approaching the discontinuity from one side. Appl. Numer. Math. 67, 98–110 (2013)CrossRefMathSciNetMATH

11.

Esposito, J.M., Kuman, V.: A state event detection algorithm for numerically simulating hybrid systems with model singularities. ACM Trans. Model. Comput. Simul. 17(1), 1–22 (2007)CrossRef

12.

Feng, Z., Utkin, V.I.: Adaptive simulation and control of variable-structure control systems in sliding regimes. Automatica 32, 1037–1042 (1996)CrossRefMATHMathSciNet

13.

Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and Its Applications, Kluwer Academic, Dordrecht (1988)CrossRefMATH

14.

Guglielmi, N., Hairer, E.: Computing breaking points in implicit delay differential equations. Adv. Comput. Math. 29, 229–247 (2008)CrossRefMathSciNetMATH

15.

Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: structure-preserving algorithms for ordinary differential equations. Springer, Berlin (2006)MATH

16.

Li, Y., Wu, H.: Global stability analysis for periodic solution in discontinuous neural networks with nonlinear growth activations. Adv. Differ. Equ. 2009, 798685 (2009). doi:10.1155/2009/798685

17.

Montijano, J.L.: Runge-Kutta methods for the numerical solution of discontinuous systems of Filippov type. Talk given at Scicade Conference, Valladolid Spain (2013)

18.

Najaf, M., Nikoukhah, R.: Modeling and simulation of differential equations in Scicos. Modelica, The Modelica Association pp. 177–185 (2006)

19.

Shampine, L.F.: Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. B 12, 1287–1296 (1986)CrossRefMathSciNetMATH

20.

Shampine, L.F., Thompson, S.: Event location for ordinary differential equations. Comput. Math. Appl. 39, 43–54 (2000)CrossRefMathSciNetMATH

21.

Zhihong, M., Yu, X.H.: Terminal sliding mode control of MIMO linear systems. IEEE Trans. Circuits Syst. I 44(11), 1065–1070 (1997)CrossRefMathSciNet

Titel: One-sided direct event location techniques in the numerical solution of discontinuous differential systems
verfasst von: Luca Dieci
Luciano Lopez
Publikationsdatum: 01.12.2015
Verlag: Springer Netherlands
Erschienen in: BIT Numerical Mathematics / Ausgabe 4/2015
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-014-0538-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Weitere Artikel der Ausgabe 4/2015

An optimal a priori error estimate in the maximum norm for the Il’in scheme in 2D

An efficient collocation method for a Caputo two-point boundary value problem

On the relation between the randomized extended Kaczmarz algorithm and coordinate descent

A compact finite difference method for solving a class of time fractional convection-subdiffusion equations

A posteriori error analysis for finite element methods with projection operators as applied to explicit time integration techniques

Spectral method for solving high order nonlinear boundary value problems via operational matrices