Fast algorithms for high-order numerical methods for space-fractional diffusion equations

Parallel-in-time multigrid for space–time finite element approximations of two-dimensional space-fractional diffusion equations

“…u(x,y,t) = 0, t ∈ I, (x,y) ∈ ∂Ω (1.2) u(x,y,0) = ψ 0 (x,y), (x,y) ∈ Ω (1.3) with orders 1/2 < β, γ < 1, constants K x , K y > 0, solution domain Ω = (a,b)×(c,d), and Riesz fractional derivatives Since closed-form analytical solutions of fractional models are rarely accessible in practice, the numerical solutions become very prevalent to empower their successful applications. In literatures, numerical methods of SFDEs proposed to achieve high accuracy and efficiency include finite difference [3][4][5][6][7][8][9][10], finite element (FE) [11][12][13][14][15][16], finite volume [17][18][19] and spectral (element) [20][21][22] methods. It must be emphasized that no matter which discretization is applied, there usually persists intensive computational task in nonlocality caused by fractional differential operators [23].…”

Fast solution algorithms for exponentially tempered fractional diffusion equations

“…The proposed fast‐direct method is based on expressing the inverse of the Toeplitz matrix as a combination of multiplications of skew‐circulant and circulant matrices , and then the linear system is solved by direct multiplications of circulant and skew‐circulant matrices with certain vectors in each time step. Such multiplications can be done via a fixed number of fast Fourier transform (FFT) in operations.…”

High‐order compact schemes for fractional differential equations with mixed derivatives

“…For practical applications, one needs to find numerical solutions to fractional differential equations. Extensive study on efficient numerical methods for these equations have been done, see [3][4][5][6][7][8][9][10][11][12] and the references therein. In this article, we study high-order compact schemes.…”