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.

2021/07/20