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A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains

Rafael Bailo, José A. Carrillo, Stefano Fronzoni, David Gómez-Castro

TL;DR

The paper develops a finite-volume framework for a new fractional Laplacian on bounded domains, formulated as a conservation law to naturally impose no-flux boundaries and enable robust numerical treatment of nonlocal diffusion. It constructs the operator via a bounded-domain inversion strategy and proves well-posedness in the β=0 case using Hille–Yosida theory, while also formulating a practical, implicit time-stepping scheme. The numerical method, designed for 1D and extended to 2D via dimensional splitting, accurately captures fractional diffusion and Lévy-Fokker-Planck dynamics, with extensive validation against analytical solutions and analysis of boundary behavior, steady states, and long-time asymptotics. Collectively, the work provides a scalable, structure-preserving approach for simulating nonlocal diffusion in bounded domains, with insights into stationary states and domain-size effects that are relevant for applications in physics and biology.

Abstract

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the Lévy-Fokker-Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.

A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains

TL;DR

The paper develops a finite-volume framework for a new fractional Laplacian on bounded domains, formulated as a conservation law to naturally impose no-flux boundaries and enable robust numerical treatment of nonlocal diffusion. It constructs the operator via a bounded-domain inversion strategy and proves well-posedness in the β=0 case using Hille–Yosida theory, while also formulating a practical, implicit time-stepping scheme. The numerical method, designed for 1D and extended to 2D via dimensional splitting, accurately captures fractional diffusion and Lévy-Fokker-Planck dynamics, with extensive validation against analytical solutions and analysis of boundary behavior, steady states, and long-time asymptotics. Collectively, the work provides a scalable, structure-preserving approach for simulating nonlocal diffusion in bounded domains, with insights into stationary states and domain-size effects that are relevant for applications in physics and biology.

Abstract

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the Lévy-Fokker-Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.
Paper Structure (16 sections, 1 theorem, 43 equations, 15 figures)

This paper contains 16 sections, 1 theorem, 43 equations, 15 figures.

Table of Contents

  1. Introduction
  2. A new fractional Laplacian in bounded domains
  3. Numerical schemes
  4. One dimension
  5. Two dimensions
  6. Dimensional Splitting
  7. Numerical experiments
  8. One dimension
  9. Fractional diffusion
  10. Singular behaviour at the boundary
  11. Steady states as a function of domain size
  12. Convergence of steady states
  13. Two dimensions
  14. Steady states as a function of domain size
  15. Convergence of steady states
  16. ...and 1 more sections

Key Result

Theorem 2.1

There exists a unique semigroup $\mathcal{S}(t) : L^2 (\Omega) \to L^2 (\Omega)$ of solutions of eq:heat. In fact, if $\mathcal{B} u_0 \in H^2(\Omega)$ then $\mathcal{B} u(t) \in H^2 (\Omega)$ for all times and the equation is satisfied in the operator sense.

Figures (15)

  • Figure 1: Dimensional splitting, row update. The split implicit problem considers information on the whole domain, but the density is allowed to change only within a single row. These updates take place independently for each row in parallel, and can be parallelised.
  • Figure 2: Fractional heat equation \ref{['eq:fracHE']} in one dimension. Numerical solution $\bm{\bar{\rho}}^{m}$ on $\Omega=(-R,R)$ and explicit solution $\phi$ on ${\mathbb{R}}$. Scheme \ref{['eq:scheme1D']}, $\alpha = 1 + \varepsilon$, $R=100$, $\Delta x=0.1$, $\Delta t=0.1$. Good agreement is shown on the interior of the domain; boundary effects are visible.
  • Figure 3: Fractional heat equation \ref{['eq:fracHE']} in one dimension. Numerical solution $\bm{\bar{\rho}}^{m}$ and explicit steady state $\rho_\infty$ on $\Omega=(-R,R)$. Scheme \ref{['eq:scheme1D']}, $\alpha = 1.5$, $R=50$, $\Delta x=0.1$, $\Delta t=0.5$. The numerical solution tends to $\rho_\infty$.
  • Figure 4: Lévy-Fokker-Planck equation \ref{['eq:fracFP']} in one dimension. Numerical solution $\bm{\bar{\rho}}^{m}$, exact solution $\rho^*$, and explicit steady state $\rho_\infty$. Scheme \ref{['eq:scheme1D']}, $\alpha = 1+\varepsilon$, $R=50$, $\Delta x=0.05$, $\Delta t=0.01$. The numerical solution clearly tends to $\rho_\infty$.
  • Figure 5: Lévy-Fokker-Planck equation \ref{['eq:fracFP']} in one dimension. ${L^{1}}(\Omega)$ distance between the numerical steady state with $\alpha=1+\varepsilon$ on $\Omega=(-R,R)$ and the explicit steady state with $\alpha=1$. Scheme \ref{['eq:scheme1D']}, $\Delta x=2R/2^{12}$, $\Delta t=0.1$. The mismatch decreases as $R$ increases.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Theorem 2.1
  • proof