Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation

  • In this article we analyse a generalized trapezoidal rule for initial value problems with piecewise smooth right-hand side F : IR^n -> IR^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.

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Author:Andreas GriewankORCiD, Richard HasenfelderORCiD, Manuel RadonsORCiD, Lutz Lehmann, Tom StreubelORCiD
Document Type:Article
Parent Title (English):Optimization Methods and Software
First Page:1089
Last Page:1107
Publisher:Taylor & Francis
Year of first publication:2018