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Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation. (English) Zbl 1401.65070
Summary: In this article we analyse a generalized trapezoidal rule for initial value problems with piecewise smooth right-hand side \(F:\mathbb R^n\to \mathbb R^n\) 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.

MSC:
65L05 Numerical methods for initial value problems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
65L99 Numerical methods for ordinary differential equations
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
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