Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation. (2nd November 2018) - Griewank, Andreas; Hasenfelder, Richard; Radons, Manuel; Lehmann, Lutz; Streubel, Tom
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.