Fangying Song

Tingting Wang, Fangying Song, Hong Wang et al.

The Gray-Scott (GS) model represents the dynamics and steady state pattern formation in reaction-diffusion systems and has been extensively studied in the past. In this paper, we consider the effects of anomalous diffusion on pattern formation by introducing the fractional Laplacian into the GS model. First, we prove that the continuous solutions of the fractional GS model are unique. We then introduce the Crank-Nicolson (C-N) scheme for time discretization and weighted shifted Grünwald difference operator for spatial discretization. We perform stability analysis for the time semi-discrete numerical scheme, and furthermore, we analyze numerically the errors with benchmark solutions that show second-order convergence both in time and space. We also employ the spectral collocation method in space and C-N scheme in time to solve the GS model in order to verify the accuracy of our numerical solutions. We observe the formation of different patterns at different values of the fractional order, which are quite different than the patterns of the corresponding integer-order GS model, and quantify them by using the radial distribution function (RDF). Finally, we discover the scaling law for steady patterns of the RDFs in terms of the fractional order $1<α\leq 2 $.

Xianjuan Li, Zhiping Mao, Fangying Song et al.

We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the singular kernel with a degenerate kernel, we construct a hierarchical matrix (H-matrix) to represent the stiffness matrix of the SEM and provide error estimates verified numerically. We can solve efficiently the H-matrix approximation problem using a hierarchical LU decomposition method, which reduces the computational cost to $O(R^2 N_d \log^2N) +O(R^3 N_d \log N)$, where $R$ it is the rank of submatrices of the H-matrix approximation, $N_d$ is the total number of degrees of freedom and $N$ is the number of elements. However, we lose the high accuracy of the SEM. Thus, we solve the corresponding preconditioned system by using the H-matrix approximation problem as a preconditioner, recovering the high order accuracy of the SEM. The condition number of the preconditioned system is independent of the polynomial degree $P$ and grows with the number of elements, but at modest values of the rank $R$ is below order 10 in our experiments, which represents a reduction of more than 11 orders of magnitude from the unpreconditioned system; this reduction is higher in the two-sided fractional derivative compared to one-sided fractional derivative. The corresponding cost is $O(R^2 N_d \log^2 N)+O(R^3 N_d \log N)+O(N_d^2)$. Moreover, by using a structured mesh (uniform or geometric mesh), we can further reduce the computational cost to $O(R^2 N_d\log^2 N) +O(R^3 N_d \log N)+ O(P^2 N\log N)$ for the preconditioned system. We present several numerical tests to illustrate the proposed algorithm using $h$ and $p$ refinements.