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.
| 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 |
