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Fractional variational integrators based on convolution quadrature

Khaled Hariz, Fernando Jiménez, Sina Ober-Blöbaum

TL;DR

This paper addresses the numerical integration of dissipative systems with fractional damping by formulating a variational problem via a restricted Hamilton's principle that doubles the state and employs fractional derivatives. It develops higher-order fractional variational integrators (FVIs) by coupling high-order variational discretizations for the conservative part with convolution quadrature (CQ) for the fractional damping, specifically using backward difference formulas (BDF) to generate CQ weights. The authors prove discrete Euler–Lagrange equations in both continuous and CQ-based discrete settings, analyze convergence and saturation of CQ, and demonstrate through damped oscillator and Bagley–Torvik tests that the method preserves dissipation structure and achieves order up to 2 in many cases, with potential order improvements requiring corrections or alternative CQ strategies. The work establishes a framework for structure-preserving, high-order time integration of non-local fractional dissipative systems, with practical implications for accurate long-time simulations in mechanics and damping phenomena. It also outlines limitations (e.g., saturation and inner-node handling) and points to future directions such as Runge–Kutta CQ and correction terms to recover higher-order convergence.

Abstract

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.

Fractional variational integrators based on convolution quadrature

TL;DR

This paper addresses the numerical integration of dissipative systems with fractional damping by formulating a variational problem via a restricted Hamilton's principle that doubles the state and employs fractional derivatives. It develops higher-order fractional variational integrators (FVIs) by coupling high-order variational discretizations for the conservative part with convolution quadrature (CQ) for the fractional damping, specifically using backward difference formulas (BDF) to generate CQ weights. The authors prove discrete Euler–Lagrange equations in both continuous and CQ-based discrete settings, analyze convergence and saturation of CQ, and demonstrate through damped oscillator and Bagley–Torvik tests that the method preserves dissipation structure and achieves order up to 2 in many cases, with potential order improvements requiring corrections or alternative CQ strategies. The work establishes a framework for structure-preserving, high-order time integration of non-local fractional dissipative systems, with practical implications for accurate long-time simulations in mechanics and damping phenomena. It also outlines limitations (e.g., saturation and inner-node handling) and points to future directions such as Runge–Kutta CQ and correction terms to recover higher-order convergence.

Abstract

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.
Paper Structure (20 sections, 6 theorems, 69 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 20 sections, 6 theorems, 69 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Higher order discrete variational mechanics
  3. Hamilton's principle and Euler-Lagrange equations
  4. Discrete Hamilton's principle and discrete Euler-Lagrange equations
  5. Higher order approximations of the action
  6. (1) Trajectories space
  7. (2) Quadrature for the action integral
  8. Fractional integrals, fractional derivatives and their approximations by discrete convolutions
  9. Fractional integrals and fractional derivatives
  10. Fractional integrals
  11. Fractional derivatives
  12. Fractional integrals as convolutions
  13. Discrete convolution
  14. Restricted variational principle for the dynamics of Lagrangian systems subject to fractional damping
  15. Continuous setting
  16. ...and 5 more sections

Key Result

Lemma 3.1

Consider two discrete series $\left\{ f_k\right\}_{\tiny 0:N}, \left\{ g_k\right\}_{\tiny 0:N}$. Then the following properties hold true:

Figures (8)

  • Figure 1: Polynomial interpolation principles. On each subinterval $[t_k,t_{k+1}]$, the trajectory interpolated by a polynomial passing through the points $\{q_k^i\}_{i=0}^{s}$ associated with the control points $\{d_\nu h\}$. The evaluations are made by the quadrature points for the $\{c_ih\}$, indicated by cross points.
  • Figure 2: Log-Log plot the error $e(h)$ versus $h$ corresponds to the Caputo fractional derivative of order $1/2$ ($\alpha=-1/2$ in \ref{['eq:caputoerror']}). As expected, the convergence starts to saturate at $p=1$ (upper-left), $p=3$ (upper-right), $p=4$ (lower-left) and $p=5$ (lower-right).
  • Figure 3: Damped harmonic oscillator \ref{['eq:damped-oscillator']} ($\alpha=1/2$). Left: Exact solution vs FVI-BDF1CQ method for $h=0.125$. Right: Energy behaviour for $h=0.125$.
  • Figure 4: Damped harmonic oscillator \ref{['eq:damped-oscillator']}. Left: absolute errors for $h=0.125$. Right: Log-Log plot of the global error presented on $t\in [0,16]$ for $h =16/2^i,\ i=4,\ldots,11$.
  • Figure 5: Bagley-Torvik equation \ref{['eq:torvik']}. Log-Log plot of the global errors on $t\in [0,1]$ for $h = 1/2^i,\ i=1,\ldots,8$. Left: case \ref{['torvik1']}. Right: case \ref{['torvik2']}.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Remark 3.1
  • Lemma 3.1
  • proof
  • Definition 3.1
  • Theorem 3.1: Theorem 2.6 in Lubich1
  • Theorem 4.1
  • Theorem 4.2
  • Lemma 5.1
  • proof
  • Theorem 5.1
  • ...and 2 more