Fractional Variational Integrators Based on Convolution Quadrature

Abstract Fractional dissipation is a powerful tool to study nonlocal physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. In Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021), a new approach is proposed to deal with dissipative systems including fractionally damped systems in a variational way for both, the continuous and discrete setting. It is based on the doubling of variables and their fractional derivatives. The aim of this work is to derive higher-order fractional variational integrators by means of convolution quadrature (CQ) based on backward difference formulas. We then provide numerical methods that are of order 2 improving a previous result in Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021). The convergence properties of the fractional variational integrators and saturation effects due to the approximation of the fractional derivatives by CQ are studied numerically.