TY - GEN
A1 - Griewank, Andreas
A1 - Hasenfelder, Richard
A1 - Radons, Manuel
A1 - Lehmann, Lutz
A1 - Streubel, Tom
T1 - Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation
N2 - 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.
T3 - ZIB-Report - 17-44
KW - Automatic Differentiation
KW - Lipschitz Continuity
KW - Piecewise Linearization
KW - Nonsmooth
KW - Trapezoidal Rule
KW - Energy Preservation
KW - Dense Output
Y1 - 2017
UR - https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/6463
UR - https://nbn-resolving.org/urn:nbn:de:0297-zib-64639
SN - 1438-0064
ER -