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Upwind-and-shifted numerical scheme for fractional convection equation

Lot-Kei Chou, Wan-Na Deng, Yuan-Yuan Huang, Siu-Long Lei

TL;DR

This work develops high-order, unconditionally stable finite-difference schemes for a space fractional convection equation with $α∈(0,1)$. It introduces upwind-based approximations and an upwind-and-shifted variant, achieving a spatial order of $3-α$ and proving stability through generating-function analysis that leverages the Lerch transcendent. The resulting schemes lead to Toeplitz linear systems that are efficiently solvable by GMRES, with numerical tests confirming second-order temporal and spatial accuracy and effective reproduction of Lévy-flight distributions. The methods offer robust, accurate tools for simulating anomalous transport modeled by fractional convection, with practical implications for Lévy flight simulations and related applications.

Abstract

Fundamental solution of a space fractional convection equation of order $α$ is the probability density function of Lévy flights with long-tailed $α$-stable jump length distribution. By studying an upwind second-order implicit finite difference scheme for the equation with $α\in(0,1)$, an upwind-and-shifted scheme with order $3-α$ is obtained in this paper, and the scheme is shown to be unconditionally stable for a wide range of $α$. Numerical examples, including simulations on a probability density function, are presented showing the effectiveness of the numerical schemes.

Upwind-and-shifted numerical scheme for fractional convection equation

TL;DR

This work develops high-order, unconditionally stable finite-difference schemes for a space fractional convection equation with . It introduces upwind-based approximations and an upwind-and-shifted variant, achieving a spatial order of and proving stability through generating-function analysis that leverages the Lerch transcendent. The resulting schemes lead to Toeplitz linear systems that are efficiently solvable by GMRES, with numerical tests confirming second-order temporal and spatial accuracy and effective reproduction of Lévy-flight distributions. The methods offer robust, accurate tools for simulating anomalous transport modeled by fractional convection, with practical implications for Lévy flight simulations and related applications.

Abstract

Fundamental solution of a space fractional convection equation of order is the probability density function of Lévy flights with long-tailed -stable jump length distribution. By studying an upwind second-order implicit finite difference scheme for the equation with , an upwind-and-shifted scheme with order is obtained in this paper, and the scheme is shown to be unconditionally stable for a wide range of . Numerical examples, including simulations on a probability density function, are presented showing the effectiveness of the numerical schemes.

Paper Structure

This paper contains 10 sections, 8 theorems, 43 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Approximations of fractional derivatives
  3. Upwind approximation
  4. Two-term upwind approximation
  5. Upwind-and-shift approximation
  6. Finite difference scheme and the discrete linear system
  7. Stability and convergence analysis
  8. Numerical results
  9. Conclusion
  10. Acknowledgments

Key Result

Lemma 1

For all $\alpha\in(0,1)$, we have

Figures (4)

  • Figure 1: Contour of $\textrm{Re}(g_{2}(\theta,\alpha))$ with $\alpha\in[0.01,0.99],~\theta\in[0.001,3.141]$.
  • Figure 2: Contour of $\textrm{Re}(g_{2}(\theta,\alpha))$ with $\alpha\in[0.98,0.999],~\theta\in[0.001,3.141]$.
  • Figure 3: Contour of $\textrm{Re}(g_{3}(\theta,\alpha))$ with $\alpha\in[0.01,0.99],~\theta\in[0.001,3.141]$.
  • Figure 4: Contour of $\textrm{Re}(g_{3}(\theta,\alpha))$ with $\alpha\in[0.99,0.999],~\theta\in[0.001,3.141]$.

Theorems & Definitions (13)

  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2
  • Proposition 3
  • Remark 3
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • ...and 3 more