Title: Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation,
Abstract: In this article we analyze a generalized trapezoidal rule for initial value
problems with piecewise smooth right hand side \(F:\R^n\to\R^n\). 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.