Fractional dissipation is a powerful tool to study non-local 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 [19], 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 [19]. The convergence properties of the fractional variational integrators and saturation effects due to the approximation of the fractional derivatives by CQ are studied numerically.

翻译：分数阶耗散是研究非局部物理现象（如阻尼模型）的有力工具。对此类系统进行数值模拟时，设计几何型（特别是变分型）积分器需要基于模型的变分公式化。文献[19]提出了一种新方法，以变分方式处理包括分数阶阻尼系统在内的耗散系统，涵盖连续与离散两种情形。该方法基于变量及其分数阶导数的增广。本文旨在通过基于后向差分公式的卷积积分（CQ）推导高阶分数阶变分积分器。我们进而提供了阶数为2的数值方法，改进了文献[19]的先前结果。通过数值方法研究了分数阶变分积分器的收敛性以及CQ近似分数阶导数导致的饱和效应。