Finite-Volume Scheme for Fractional Diffusion on Bounded Domains

Core Concepts

A new finite-volume numerical scheme is proposed to approximate the solution of the non-local diffusion problem given by the fractional heat equation and the related Lévy-Fokker-Planck equation on bounded domains.

Abstract

The authors present a new finite-volume numerical scheme to approximate the solution of the fractional heat equation and the Lévy-Fokker-Planck equation on bounded domains.
Key highlights:

The scheme is designed to handle the fractional Laplacian term by expressing it as a conservation law, which is well-suited for finite-volume discretization.
The scheme allows for the direct prescription of no-flux boundary conditions.
The authors first establish the well-posedness theory for the fractional heat equation on bounded domains.
The numerical scheme is developed in one and two dimensions, with the two-dimensional case employing a dimensional splitting approach for computational efficiency.
The numerical solutions are benchmarked against known analytical solutions for the Lévy-Fokker-Planck equation.
The authors explore the properties of these equations, such as their stationary states and long-time asymptotics, through numerical experiments.

Stats

The authors provide the following key figures and metrics to support their work:

The inverse Fourier transform of the fractional Laplacian symbol |ξ|^α yields the Riesz or singular integral definition of the fractional Laplacian.
The constant C(d, α) that arises in the computation of the inverse transform of |ξ|^α is given by 2^α Γ((d+α)/2) / (π^(d/2) Γ(-α/2)).
The authors derive the formulation of the Lévy-Fokker-Planck equation in divergence form, which involves the inverse fractional Laplacian operator (−Δ)^(-α/2).

Quotes

"Our approach permits the direct prescription of no-flux boundary conditions."
"Finite volume schemes have been used with success to produce structure preserving schemes for equations in divergence form of gradient flow type and related systems."
"We also highlight several spectral methods which deal exclusively with problems on unbounded domains."

Key Insights Distilled From

A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains

by Rafa... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2309.08283.pdf

Deeper Inquiries

How can the proposed finite-volume scheme be extended to handle more general boundary conditions, such as Dirichlet or Robin conditions, on bounded domains

To extend the proposed finite-volume scheme to handle more general boundary conditions on bounded domains, such as Dirichlet or Robin conditions, we can modify the flux computation at the boundaries.
For Dirichlet conditions, where the value of the solution is specified on the boundary, we can adjust the flux calculation to incorporate this information. At Dirichlet boundaries, the flux should be set to zero in the normal direction to the boundary, ensuring that the solution does not flow across the boundary. This adjustment can be implemented in the flux calculation step of the numerical scheme.
For Robin conditions, which involve a combination of Dirichlet and Neumann conditions, the flux calculation needs to consider both the value of the solution and its derivative at the boundary. By incorporating the Robin condition into the flux calculation, we can ensure that the numerical scheme respects the specified boundary conditions.
In summary, extending the finite-volume scheme to handle more general boundary conditions involves adapting the flux calculation to incorporate the specific boundary conditions at Dirichlet and Robin boundaries. This modification ensures that the numerical scheme accurately captures the behavior of the solution at the boundaries of the domain.

What are the potential challenges and limitations of the dimensional splitting approach used in the two-dimensional case, and how could it be further improved

The dimensional splitting approach used in the two-dimensional case has certain challenges and limitations that need to be considered:

Accuracy: Dimensional splitting may introduce errors due to the sequential nature of the updates along each dimension. The accuracy of the overall solution depends on the convergence of the individual updates and their interaction.

Boundary Effects: Dimensional splitting may not capture boundary effects accurately, especially when the solution exhibits complex behavior near the boundaries. The sequential updates along each dimension may not fully account for the interactions at the boundaries.

Computational Efficiency: While dimensional splitting can be parallelized for row-wise or column-wise updates, the memory requirements and computational complexity may increase significantly, especially for dense matrices in higher dimensions.

To improve the dimensional splitting approach, one could consider:

Higher-order Splitting Schemes: Using higher-order splitting schemes can improve accuracy by reducing errors introduced by the sequential updates.
Adaptive Mesh Refinement: Implementing adaptive mesh refinement techniques can help focus computational resources where they are most needed, improving accuracy near boundaries.
Boundary Treatment: Developing specialized boundary treatment methods within the dimensional splitting framework to better capture boundary effects.
By addressing these challenges and incorporating improvements, the dimensional splitting approach can be further enhanced for more accurate and efficient numerical simulations in two-dimensional cases.

Can the proposed numerical framework be adapted to handle other types of non-local diffusion operators, such as those arising in anomalous diffusion models in biology or finance

The proposed numerical framework can be adapted to handle other types of non-local diffusion operators, such as those arising in anomalous diffusion models in biology or finance, by modifying the discretization of the fractional operators in the finite-volume scheme.

Fractional Operators: Different non-local diffusion models may involve fractional operators with varying properties. By adjusting the discretization of these operators in the numerical scheme, the framework can accommodate the specific characteristics of the non-local diffusion model.

Anomalous Diffusion Models: Anomalous diffusion models often involve non-Gaussian behavior and long-range interactions. The numerical framework can be tailored to capture these features by incorporating appropriate numerical approximations for the non-local interactions.

Model Validation: Adapting the framework to handle different non-local diffusion operators requires validation against analytical solutions or experimental data specific to the anomalous diffusion model under consideration. This validation ensures the accuracy and reliability of the numerical results.

By customizing the numerical framework to suit the requirements of specific anomalous diffusion models, researchers can effectively simulate and analyze complex diffusion phenomena in various fields, including biology and finance.