Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation. (2nd November 2018)

• Record Type: Journal Article
• Authors: Griewank, Andreas; Hasenfelder, Richard; Radons, Manuel; Lehmann, Lutz; Streubel, Tom
• Is Part Of: Optimization methods and software. Volume 33:Number 4/6(2018)
• Page Start: 1089
• Page End: 1107
• Publication Date: 2018-11-02
• Subjects: Automatic differentiation -- Lipschitz continuity -- piecewise linearization -- nonsmooth -- Trapezoidal rule -- energy preservation -- dense output; 65L05 -- 65L06 -- 65L70 -- 65L99 -- 65P10; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2017.1378653
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : In this article we analyse a generalized trapezoidal rule for initial value problems with piecewise smooth right-hand side 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.