Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6929203 | Journal of Computational Physics | 2018 | 28 Pages |
Abstract
In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (âÎ)α/2 in hypersingular integral form. By introducing a splitting parameter, we formulate the fractional Laplacian as the weighted integral of a weak singular function, which is then approximated by the weighted trapezoidal rule. Compared to other existing methods, our method is more accurate and simpler to implement, and moreover it closely resembles the central difference scheme for the classical Laplace operator. We prove that for uâC3,α/2(R), our method has an accuracy of O(h2)uniformly for anyαâ(0,2), while for uâC1,α/2(R), the accuracy is O(h1âα/2). The convergence behavior of our method is consistent with that of the central difference approximation of the classical Laplace operator. Additionally, we apply our method to solve the fractional Poisson equation and study the convergence of its numerical solutions. The extensive numerical examples that accompany our analysis verify our results, as well as give additional insights into the convergence behavior of our method.
Related Topics
Physical Sciences and Engineering
Computer Science
Computer Science Applications
Authors
Siwei Duo, Hans Werner van Wyk, Yanzhi Zhang,