Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation

TL;DR

This paper introduces a generalized trapezoidal rule for solving initial value problems with piecewise smooth right-hand sides, improving accuracy, preserving energy in Hamiltonian systems, and extending classical methods to nonsmooth cases.

Contribution

It proposes a new generalized trapezoidal rule that enhances accuracy and energy preservation for nonsmooth dynamical systems, extending classical methods.

Findings

Achieves higher convergence order than classical methods.

Preserves energy in piecewise linear Hamiltonian systems.

Provides a third order interpolation polynomial for numerical trajectories.

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.