On Runge-Kutta convolution quadrature based fractional variational integrators

Abstract

Lagrangian systems subject to fractional damping can be incorporated into a variational framework by doubling the state variables and introducing fractional derivatives. Fractional variational integrators based on backward-differentiation convolution quadrature (BD-FCQ), combined with higher-order Galerkin methods, saturate at second-order accuracy because the multistep structure of BDFCQ does not take into account the internal stages of the Galerkin discretization. The main objective of this paper is to develop fractional variational integrators (FVIs) by combining Runge-Kutta convolution quadrature (RKCQ) for the approximation of fractional derivatives with higher-order Galerkin methods. The RKCQ approach is naturally compatible with such stage-based discretizations and is therefore better suited for the construction of higher-order schemes. We are particularly interested in the CQ based on Lobatto IIIC. Preservation properties such as energy decay, as well as convergence properties, are investigated numerically and proved for second-order schemes. The presented schemes reach 2nd, 4th and 6th order of accuracy. A brief discussion on the midpoint fractional integrator is also included.