Fangying Song

Spectral direction splitting methods for two-dimensional space fractional diffusion equations

A numerical method for a kind of time-dependent two-dimensional two-sided space fractional diffusion equations is developed in this paper. The proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization. The direction splitting approach renders the underlying two-dimensional equation into a set of one-dimensional space fractional diffusion equations at each time step. Then these one-dimensional equations are solved by using the spectral method based on weak formulations. A time error estimate is derived for the semi-discrete solution, and the unconditional stability of the fully discretized scheme is proved. Some numerical examples are presented to validate the proposed method.

Computing Fractional Laplacians on Complex-Geometry Domains: Algorithms and Simulations

We consider a fractional Laplacian defined in bounded domains by the eigen-decomposition of the integer-order Laplacian, and demonstrate how to compute very accurately (using the spectral element method) the eigenspectrum and corresponding eigenfunctions in two-dimensional prototype complex-geometry domains. We then employ these eigenfunctions as trial and test bases to first solve the fractional diffusion equation, and subsequently to simulate two-phase flow based on the Navier--Stokes equations combined with a fractional Allen--Cahn mass-preserving model. A key point to the effectiveness of an exponential convergence of this approach is the use of a weighted Gram--Schmidt orthonormalization of the eigenfunctions that guarantees accurate projection and recovery of spectral accuracy for smooth solutions. We demonstrate that even when only part of the eigenspectrum is computed accurately we can still obtain exponential convergence if we employ the complete set of the eigenvectors of the discrete Laplacian....