Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation
This work addresses the loss of accuracy in numerical integration of piecewise smooth systems, which is relevant for simulations in mechanics, control, and other domains with nonsmooth dynamics.
The authors propose a generalized trapezoidal rule for integrating piecewise smooth dynamical systems, achieving higher convergence order than the classical method while preserving energy for piecewise linear Hamiltonian systems. Numerical results demonstrate improved accuracy.
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.