Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation

Please always quote using this URN: urn:nbn:de:0297-zib-64639
  • In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side \(F:R^n \to R^n\) based on a generalization of algorithmic differentiation. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of \(F\). The advantage of the proposed generalized trapezoidal rule is threefold: Firstly, we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third order interpolation polynomial for the numerical trajectory. In the smooth case the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.

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2021/07/20

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Metadaten
Author:Andreas GriewankORCiD, Richard HasenfelderORCiD, Manuel RadonsORCiD, Lutz Lehmann, Tom StreubelORCiD
Document Type:ZIB-Report
Tag:Automatic Differentiation; Dense Output; Energy Preservation; Lipschitz Continuity; Nonsmooth; Piecewise Linearization; Trapezoidal Rule
MSC-Classification:65-XX NUMERICAL ANALYSIS
Date of first Publication:2017/07/20
Series (Serial Number):ZIB-Report (17-44)
ISSN:1438-0064
Published in:published at Optimization Methods and Software
DOI:https://doi.org/10.1080/10556788.2017.1378653