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On Runge-Kutta convolution quadrature based fractional variational integrators

Khaled Hariz, Sina Ober-Blöbaum, Fernando Jimenez

TL;DR

This work addresses variational integration for Lagrangian systems with fractional damping by using Runge-Kutta convolution quadrature (RKCQ) to approximate the fractional derivatives within a high-order Galerkin discretization framework. It introduces fractional variational integrators based on Lobatto IIIC Runge-Kutta methods, establishes semigroup and asymmetric integration-by-parts properties, and proves second-order accuracy for quadratic Lagrangians while achieving higher apparent orders ($2$, $4$, $6$) in practice, including a midpoint variant for $O(2)$ accuracy. Numerical experiments on coupled oscillators with fractional damping and the Bagley–Torvik model demonstrate energy decay preservation and convergence consistent with the theory, validating the FVIs’ stability and accuracy. The approach overcomes saturation limitations of BDF-based CQ, offers higher-order fractional integration, and provides a framework for further fractional variational error analysis and Gauss-type RKCQ developments.

Abstract

Lagrangian systems subject to fractional damping can be incorporated into a variational formalism. The construction can be made by doubling the state variables and introducing fractional derivatives \cite{JiOb2}. The main objective of this paper is to use the Runge-Kutta convolution quadrature (RKCQ) method for approximating fractional derivatives, combined with higher order Galerkin methods in order to derive fractional variational integrators (FVIs). We are specially 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 2nd order schemes. The presented schemes reach 2nd, 4th and 6th accuracy order. A brief discussion on the midpoint fractional integrator is also included.

On Runge-Kutta convolution quadrature based fractional variational integrators

TL;DR

This work addresses variational integration for Lagrangian systems with fractional damping by using Runge-Kutta convolution quadrature (RKCQ) to approximate the fractional derivatives within a high-order Galerkin discretization framework. It introduces fractional variational integrators based on Lobatto IIIC Runge-Kutta methods, establishes semigroup and asymmetric integration-by-parts properties, and proves second-order accuracy for quadratic Lagrangians while achieving higher apparent orders (, , ) in practice, including a midpoint variant for accuracy. Numerical experiments on coupled oscillators with fractional damping and the Bagley–Torvik model demonstrate energy decay preservation and convergence consistent with the theory, validating the FVIs’ stability and accuracy. The approach overcomes saturation limitations of BDF-based CQ, offers higher-order fractional integration, and provides a framework for further fractional variational error analysis and Gauss-type RKCQ developments.

Abstract

Lagrangian systems subject to fractional damping can be incorporated into a variational formalism. The construction can be made by doubling the state variables and introducing fractional derivatives \cite{JiOb2}. The main objective of this paper is to use the Runge-Kutta convolution quadrature (RKCQ) method for approximating fractional derivatives, combined with higher order Galerkin methods in order to derive fractional variational integrators (FVIs). We are specially 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 2nd order schemes. The presented schemes reach 2nd, 4th and 6th accuracy order. A brief discussion on the midpoint fractional integrator is also included.

Paper Structure

This paper contains 22 sections, 7 theorems, 76 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Structure of the paper
  4. Discrete Lagrangian Mechanics
  5. Lagrangian Hamilton's Principle
  6. Discrete Lagrangian Hamilton's Principle
  7. Galerkin Variational Methods
  8. (1) Trajectories space
  9. (2) Quadrature for the action integral
  10. Fractional Operators and their Approximations
  11. Fractional Integrals and Fractional Derivatives
  12. Runge-Kutta Convolution Quadratures
  13. Runge-Kutta methods
  14. Runge-Kutta Convolution Quadratures
  15. Fractional Variational Integrators with RKCQ
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let $K^{(\alpha)}$ be the Laplace transform of the kernel which is analytic in the half-plane $\mathrm{Re} s > \sigma > 0$, such that for some real exponent $\mu$ and bounding factor $C_K(\sigma)> 0$, the operator norm is bounded as follows: Let a Runge-Kutta method which satisfies Assumption assumption, $r> \max(p+\mu+1,p,q+1)$ and $f\in C^r([0,T],\mathbb{R})$, then there exist $\bar{h}>0$ and $

Figures (5)

  • Figure 1: Numerical results obtained over the interval $[0,20]$ with a step size $h=0.2$ for bidimensional forced oscillator \ref{['eq:damped-oscillator']}$(\alpha=1/2)$. Top: Numerical trajectories using 2-stage Lobatto IIIC. Bottom left: Energy decay. Bottom right: Energy relative error $E_{\text{err}}(t_i)$.
  • Figure 2: Bidimensional forced oscillator \ref{['eq:damped-oscillator']}. The accuracy of FVI based on Lobatto IIIC CQ with respect to position and momentum, Algorithm \ref{['alg:FractionalAlgorithm']} with $t\in[0,30]$ and $N=2^i,\ i=5,\ldots,12$.
  • Figure 3: Fractional Bagley–Torvik equation \ref{['eq:Torfik']} with $\alpha=1/4$, i.e., involves the half-derivative. Left: Numerical trajectory. Right: The accuracy of Lobatto FVI, Algorithm \ref{['alg:FractionalAlgorithm2']} with $t \in [0, 1]$ and $N = 2^i,\ i = 2, \ldots, 8.$
  • Figure 4: 1D damped oscillator \ref{['eq:1DdampedOsci']}. Left: Numerical trajectory. Right: The accuracy of FVI with MIDCQ, Algorithm \ref{['alg:FractionalAlgorithm2']} with $t \in [0, 16]$ and $N = 2^i,\ i = 4, \ldots, 11.$
  • Figure 5: Fractional Bagley–Torvik equation \ref{['eq:Torfik']}$(\alpha=1/4)$. Left: Numerical trajectory. Right: The accuracy of FVI with MIDCQ, Algorithm \ref{['alg:FractionalAlgorithm2']} with $t \in [0, 1]$ and $N = 2^i,\ i = 4, \ldots, 11.$

Theorems & Definitions (17)

  • Definition 3.1: Fractional integrals
  • Definition 3.2: Fractional derivatives
  • Remark 3.1
  • Theorem 3.1
  • Theorem 4.1: FEL equations
  • Remark 4.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 7 more