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Fractional material derivative: pointwise representation and a finite volume numerical scheme

Łukasz Płociniczak, Marek A. Teuerle

TL;DR

This work analyzes the fractional material derivative (FMD), a nonlocal operator arising in Lévy-walk scaling limits. It derives a local pointwise representation that shows the FMD acts as a Riemann-Liouville derivative along the characteristic direction, enabling definition on locally integrable spaces and facilitating explicit solvability results. A conservative finite-volume upwind scheme with an L1 time discretization is developed, and the authors prove stability, convergence, and probability conservation, complemented by numerical illustrations. The results yield efficient, deterministic computation of Lévy-walk PDFs with potential speedups over Monte Carlo methods, highlighting both theoretical insights and practical numerical approaches for anomalous transport models.

Abstract

The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of Lévy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier-Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation in time and space, pointwise, of the fractional material derivative. This allows us to define it on a space of locally integrable functions which is larger than the original one in which Fourier and Laplace transform exist as functions. We consider several typical differential equations involving the fractional material derivative and provide conditions for their solutions to exist. In some cases, the analytical solution can be found. For the general initial value problem, we devise a finite volume method and prove its stability, convergence, and conservation of probability. Numerical illustrations verify our analytical findings. Moreover, our numerical experiments show superiority in the computation time of the proposed numerical scheme over a Monte Carlo method applied to the problem of probability density function's derivation.

Fractional material derivative: pointwise representation and a finite volume numerical scheme

TL;DR

This work analyzes the fractional material derivative (FMD), a nonlocal operator arising in Lévy-walk scaling limits. It derives a local pointwise representation that shows the FMD acts as a Riemann-Liouville derivative along the characteristic direction, enabling definition on locally integrable spaces and facilitating explicit solvability results. A conservative finite-volume upwind scheme with an L1 time discretization is developed, and the authors prove stability, convergence, and probability conservation, complemented by numerical illustrations. The results yield efficient, deterministic computation of Lévy-walk PDFs with potential speedups over Monte Carlo methods, highlighting both theoretical insights and practical numerical approaches for anomalous transport models.

Abstract

The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of Lévy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier-Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation in time and space, pointwise, of the fractional material derivative. This allows us to define it on a space of locally integrable functions which is larger than the original one in which Fourier and Laplace transform exist as functions. We consider several typical differential equations involving the fractional material derivative and provide conditions for their solutions to exist. In some cases, the analytical solution can be found. For the general initial value problem, we devise a finite volume method and prove its stability, convergence, and conservation of probability. Numerical illustrations verify our analytical findings. Moreover, our numerical experiments show superiority in the computation time of the proposed numerical scheme over a Monte Carlo method applied to the problem of probability density function's derivation.
Paper Structure (9 sections, 6 theorems, 94 equations, 7 figures)

This paper contains 9 sections, 6 theorems, 94 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A link between fractional material derivatives and stochastic processes
  3. Integro-differential form of the fractional material derivative
  4. Finite-volume upwind method
  5. Numerical simulations and illustrations
  6. A model problem for the material derivative
  7. One-sided probability distribution function problem: a comparison with Monte Carlo approximation
  8. Conclusions
  9. Acknowledgement

Key Result

Theorem 1

Let $u$ be of $x$-Schwarz class and $t$-exponential on $\mathbb{R}\times\mathbb{R}_+$ so that the Fourier-Laplace transform is well-defined. Then, for $0<\alpha<1$ we have

Figures (7)

  • Figure 1: The sample one-dimensional trajectories of 3 types of LWs: wait-first, jump-first and continuous LW. Here, we assume that the probability of choosing the new direction being left or right at the renewals is equal to 0.5. Later on, we focus only on cases, where this probability of going left (or right) is equal to 1.
  • Figure 2: Lines of dependence of the fractional material derivative given by (\ref{['eqn:MatDerInt']}).
  • Figure 3: The $L^\infty$ norm error for $h\in {\{2^{-11},2^{-10},\ldots,2^{-4}\}}$ and selected values of $\alpha\in\{0.1,0.25,0.5,0.75,0.9\}$ for the numerical solution to \ref{['eqn:ModelProb']} with $u(x,0)=0$ and $f(x,t)=t^\mu$.
  • Figure 4: A comparison of numerical solution obtained using \ref{['eqn:NumMetModel']} and exact solution for $\alpha=0.5$ for the PDF of the scaling limit of the one-sided wait-first Lévy walk, see eq. \ref{['eqn:ULWProblem']} in Example \ref{['ex:waitFirstLW']}.
  • Figure 5: The $L^2$ norm error for $h\in {\{2^{-11},2^{-10},\ldots,2^{-4}\}}$ and selected values of $\alpha\in\{0.1,0.25,0.5,0.75,0.9\}$ for the numerical solution of the PDF problem of the scaling limit of the one-sided wait-first Lévy walk, see Theorem \ref{['thm:Density']} and eq. \ref{['eqn:ULWProblem']} in Example \ref{['ex:waitFirstLW']}.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem 1: Pointwise representation
  • proof
  • Proposition 1
  • proof
  • Theorem 2: One-sided probability density problem
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Proposition 2: Stability
  • ...and 5 more