In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian (-Δ)α/2 (0 < α < 2) in hypersingular integral form. The proposed methods provide a fractional analogue of the central difference schemes to the fractional Laplacian. As α → 2-, they collapse to the central difference schemes of the classical Laplace operator −Δ. We prove that our methods are consistent if 𝑢 ∈ C[α], α –[α]+ ε (ℝd), and the local truncation error is 𝓞 (hε), with ε > 0 a small constant and [· } denoting the floor function. If 𝑢 ∈ C2+[α], α -[ α]+ ε (ℝd), they can achieve the second order of accuracy for any α ∈ (0,2). These results hold for any dimension d ≥ 1 and thus improve the existing error estimates of the one-dimensional cases in the literature. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should at most satisfy 𝑢 ∈ C1,1 (ℝd). One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.
- Finite difference methods,
- Fractional Allan-Cahn equation,
- Fractional Poisson equation,
- Integral fractional Laplacian,
- Montgomery identity
Available at: http://works.bepress.com/yanzhi-zhang/32/