zbMATH — the first resource for mathematics

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
dcc
Full Text:
References:
 [1] Atkinson, K., An Introduction to Numerical Analysis, (1989), John Wiley & Sons, New York · Zbl 0718.65001 [2] Clarke, F., Optimization and Nonsmooth Analysis, (1983), Wiley-Interscience, New York · Zbl 0582.49001 [3] di Bernardo, M.; Budd, C.; Champneys, A.; Kowalczyk, P., Piecewise-smooth Dynamical Systems, (2008), Springer, London · Zbl 1146.37003 [4] Enright, W.; Jackson, K.; Nørsett, S.; Thomsen, P., effective solution of discontinuous ivps using a Runge–Kutta formula pair with interpolants, Appl. Math. Comput., 27, 313-335, (1988) · Zbl 0651.65058 [5] Griewank, A., on stable piecewise linearization and generalized algorithmic differentiation, Optim. Method Softw., 28, 6, 1139-1178, (2013) · Zbl 1278.65021 [6] Griewank, A.; Walther, A., Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, (2008) · Zbl 1159.65026 [7] Piecewise linear secant approximation via algorithmic piecewise differentiation, Submitted 2016 [8] Griewank, A.; Bernt, J.; Radons, M.; Streubel, T., solving piecewise linear systems in ABS-normal form, Linear Algebra Appl., 471, 500-530, (2015) · Zbl 1308.90179 [9] Hairer, E.; Nørsett, S.; Wanner, G., Solving Ordinary Differential Equations I, (1993), Springer, Berlin, Heidelberg · Zbl 0789.65048 [10] Naumann, U., The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, (2012), SIAM · Zbl 1275.65015 [11] Chartier, Philippe; Faou, Erwan, geometric integrators for piecewise smooth Hamiltonian systems, ESAIM: M2AN, 42, (2008) · Zbl 1145.65110 [12] Quispel, G. R.W.; McLaren, D., A new class of energy-preserving numerical integration methods, J. Phys. A, 41, (2008) · Zbl 1132.65065 [13] Radons, M., direct solution of piecewise linear systems, Theor. Comput. Sci., 626, 97-109, (2016) · Zbl 1336.68144 [14] Scholtes, S., Introduction to Piecewise Differentiable Equations, (2012), Springer, New York · Zbl 06046475 [15] Shampine, L.; Thomson, S., event location for ordinary differential equations, Comput. Math. Appl., 39, 43-54, (2000) · Zbl 0956.65055 [16] Streubel, T.; Griewank, A.; Radons, M.; Bernt, J., representation and analysis of piecewise linear functions in ABS-normal form, Proc. of the IFIP TC, 7, 323-332, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.