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Mathematics > Numerical Analysis

arXiv:2309.08283 (math)
[Submitted on 15 Sep 2023 (v1), last revised 16 Apr 2024 (this version, v2)]

Title:A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains

Authors:Rafael Bailo, José A. Carrillo, Stefano Fronzoni, David Gómez-Castro
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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.
Subjects: Numerical Analysis (math.NA)
MSC classes: 35R11, 65N08
Cite as: arXiv:2309.08283 [math.NA]
  (or arXiv:2309.08283v2 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2309.08283
Journal reference: European Journal of Applied Mathematics: 1-21 (2024)

Submission history

From: Stefano Fronzoni [view email]
[v1] Fri, 15 Sep 2023 09:52:47 UTC (10,636 KB)
[v2] Tue, 16 Apr 2024 12:09:43 UTC (10,577 KB)
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