Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation
- In this article we analyse a generalized trapezoidal rule for initial value problems with piecewise smooth right-hand side F : IR^n -> IR^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 |
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Document Type: | Article |
Parent Title (English): | Optimization Methods and Software |
Volume: | 33 |
First Page: | 1089 |
Last Page: | 1107 |
Publisher: | Taylor & Francis |
Year of first publication: | 2018 |
Preprint: | urn:nbn:de:0297-zib-64639 |
DOI: | https://doi.org/10.1080/10556788.2017.1378653 |