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Fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative

Jing Sun, Daxin Nie, Weihua Deng

TL;DR

The paper tackles the high storage and computation costs of convolution-quadrature methods for the Riemann-Liouville fractional derivative in two-state fractional Fokker-Planck equations. It introduces fast BE and fast SBD discretizations that represent convolution weights via geometric sequences and Gauss-Jacobi-based quadrature, enabling iterative history updates and eliminating the need for time-regularity assumptions. The authors provide rigorous error analyses showing first-order accuracy for fast BE and second-order accuracy for fast SBD, together with explicit bounds that include quadrature-error terms. Numerical experiments confirm the predicted convergence rates and demonstrate substantial speedups and memory savings, making the methods practical for large-scale, nonsmooth-time simulations of anomalous diffusion with internal states.

Abstract

Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in [Nie, Sun and Deng, arXiv: 1811.04723], which use convolution quadrature to approximate the Riemann-Liouville fractional derivative; and the schemes need huge storage and computational cost because of the non-locality of fractional derivative and the large scale of the system. This paper first provides the fast algorithms for computing the Riemann-Liouville derivative based on convolution quadrature with the generating function given by the backward Euler and second-order backward difference methods; the algorithms don't require the assumption of the regularity of the solution in time, while the computation time and the total memory requirement are greatly reduced. Then we apply the fast algorithms to solve the homogeneous fractional Fokker-Planck equations with two internal states for nonsmooth data and get the first- and second-order accuracy in time. Lastly, numerical examples are presented to verify the convergence and the effectiveness of the fast algorithms.

Fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative

TL;DR

The paper tackles the high storage and computation costs of convolution-quadrature methods for the Riemann-Liouville fractional derivative in two-state fractional Fokker-Planck equations. It introduces fast BE and fast SBD discretizations that represent convolution weights via geometric sequences and Gauss-Jacobi-based quadrature, enabling iterative history updates and eliminating the need for time-regularity assumptions. The authors provide rigorous error analyses showing first-order accuracy for fast BE and second-order accuracy for fast SBD, together with explicit bounds that include quadrature-error terms. Numerical experiments confirm the predicted convergence rates and demonstrate substantial speedups and memory savings, making the methods practical for large-scale, nonsmooth-time simulations of anomalous diffusion with internal states.

Abstract

Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in [Nie, Sun and Deng, arXiv: 1811.04723], which use convolution quadrature to approximate the Riemann-Liouville fractional derivative; and the schemes need huge storage and computational cost because of the non-locality of fractional derivative and the large scale of the system. This paper first provides the fast algorithms for computing the Riemann-Liouville derivative based on convolution quadrature with the generating function given by the backward Euler and second-order backward difference methods; the algorithms don't require the assumption of the regularity of the solution in time, while the computation time and the total memory requirement are greatly reduced. Then we apply the fast algorithms to solve the homogeneous fractional Fokker-Planck equations with two internal states for nonsmooth data and get the first- and second-order accuracy in time. Lastly, numerical examples are presented to verify the convergence and the effectiveness of the fast algorithms.

Paper Structure

This paper contains 13 sections, 12 theorems, 136 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fast evaluation of the Riemann-Liouville fractional derivative
  4. Fast BE discretization
  5. Fast SBD discretization
  6. Error analysis
  7. Error estimates for the fast BE scheme
  8. Error estimates for the fast SBD scheme
  9. Numerical experiments
  10. Performance of fast BE scheme
  11. Performance of fast SBD scheme
  12. Derivation of \ref{['equO1toderinAp']}
  13. Derivation of \ref{['equO2toderinAp']}

Key Result

Lemma 2.1

When $z\in\Sigma_{\theta,\kappa}$, $\pi/2<\theta<\pi$, and $\kappa>\max\left (2|a|^{1/\alpha_1},2|a|^{1/\alpha_2}\right )$, we have the estimates where $H(z)$, $H_{\alpha_1}(z)$ and $H_{\alpha_2}(z)$ are defined in equdefHz and H12, respectively.

Figures (2)

  • Figure 1: $\alpha=0.3$, $\tau=1/1000$, $N_{p,1}+N_{p,2}=62$
  • Figure 2: $\alpha=0.8$, $\tau=1/1000$, $N_{p,1}+N_{p,2}=82$

Theorems & Definitions (22)

  • Lemma 2.1: Nie2018
  • Lemma 2.2
  • proof
  • Lemma 3.1: Jiang2017
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Lemma 4.2: Jin2017
  • Theorem 4.3
  • Theorem 4.4
  • ...and 12 more