Local Truncation Error of Low-Order Fractional Variational Integrators

We study the local truncation error of the so-called fractional variational integrators, recently developed in [1, 2] based on previous work by Riewe and Cresson [3, 4]. These integrators are obtained through two main elements: the enlarging of the usual mechanical Lagrangian state space by the introduction of the fractional derivatives of the dynamical curves; and a discrete restricted variational principle, in the spirit of discrete mechanics and variational integrators [5]. The fractional variational integrators are designed for modelling fractional dissipative systems, which, in particular cases, reduce to mechanical systems with linear damping. All these elements are introduced in the paper. In addition, as original result, we prove (Sect. 3, Theorem 2) the order of local truncation error of the fractional variational integrators with respect to the dynamics of mechanical systems with linear damping.

[1]  C. Poole,et al.  Classical Mechanics, 3rd ed. , 2002 .

[2]  Jacky Cresson,et al.  Variational formulations of differential equations and asymmetric fractional embedding , 2012 .

[3]  P. S. Bauer.  Dissipative Dynamical Systems: I. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[4]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[5]  Riewe,et al.  Nonconservative Lagrangian and Hamiltonian mechanics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Fernando Jim'enez,et al.  A Fractional Variational Approach for Modelling Dissipative Mechanical Systems: Continuous and Discrete Settings , 2018, 1802.10544.

[7]  Sina Ober-Blöbaum,et al.  Fractional Damping Through Restricted Calculus of Variations , 2021, J. Nonlinear Sci..

[8]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[9]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[10]  Sina Ober-Blöbaum,et al.  Construction and analysis of higher order Galerkin variational integrators , 2013, Adv. Comput. Math..

[11]  George W. Patrick,et al.  Error analysis of variational integrators of unconstrained Lagrangian systems , 2008, Numerische Mathematik.

[12]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .