Finite difference methods for fractional Laplacians

Abstract

The fractional Laplacian $(-\Delta {)}^{\alpha /2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open problems in the numerical analysis of the operator. In this article, we study several different finite difference discretisations of the fractional Laplacian on uniform grids in one dimension that takes the same form. Many properties can be compared and summarised in this relatively simple setting, to tackle more important questions like the nonlocality, singularity and flat tails common in practical implementations. The accuracy and the asymptotic behaviours of the methods are also studied, together with treatment of the far field boundary conditions, providing a unified perspective on the further development of the scheme in higher dimensions.

References in corpus (5)

• Ten equivalent definitions of the fractional Laplace operator
• Fractional Laplacian in Bounded Domains
• From the long jump random walk to the fractional Laplacian
• The discontinuous Galerkin method for fractal conservation laws
• Fractional discrete Laplacian versus discretized fractional Laplacian

Cited by in corpus (5)

• A simple solver for the fractional Laplacian in multiple dimensions
• Efficient Numerical Method for Models Driven by Lévy Process via Hierarchical Matrices
• Numerical approximations for fractional elliptic equations via the method of semigroups
• Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian
• Fourier-based numerical approximation of the Weertman equation for moving dislocations