6463
eng
reportzib
0
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2017-07-20
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Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation
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.
1438-0064
urn:nbn:de:0297-zib-64639
10.1080/10556788.2017.1378653
published at Optimization Methods and Software
Andreas Griewank
Tom Streubel
Richard Hasenfelder
Manuel Radons
Lutz Lehmann
Tom Streubel
ZIB-Report
17-44
eng
uncontrolled
Automatic Differentiation
eng
uncontrolled
Lipschitz Continuity
eng
uncontrolled
Piecewise Linearization
eng
uncontrolled
Nonsmooth
eng
uncontrolled
Trapezoidal Rule
eng
uncontrolled
Energy Preservation
eng
uncontrolled
Dense Output
NUMERICAL ANALYSIS
Mathematical Optimization
MODAL-GasLab
Streubel, Tom
MODAL-Gesamt
Energy Network Optimization
https://opus4.kobv.de/opus4-zib/files/6463/ODE_paper_ZibReport.pdf