Chuanju Xu

Finite difference/spectral approximations for the time-fractional diffusion equation

In this paper, we consider the numerical resolution of a time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative (of order @a, with 0=<@a=<1). The main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time-fractional diffusion equation. The proposed method is based on a finite difference scheme in time and Legendre spectral methods in space. Stability and convergence of the method are rigourously established. We prove that the full discretization is unconditionally stable, and the numerical solution converges to the exact one with order O(@Dt^2^-^@a+N^-^m), where @Dt,N and m are the time step size, polynomial degree, and regularity of the exact solution respectively. Numerical experiments are carried out to support the theoretical claims.

A Space-Time Spectral Method for the Time Fractional Diffusion Equation

In this paper, we consider the numerical solution of the time fractional diffusion equation. Essentially, the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term. In the former case, the first-order time derivative is replaced by a fractional derivative, making the problem global in time. We propose a spectral method in both temporal and spatial discretizations for this equation. The convergence of the method is proven by providing a priori error estimate. Numerical tests are carried out to confirm the theoretical results. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the “global time dependence” can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.

Finite Difference/Spectral Approximations for the Fractional Cable Equation

Fujian NSF [S0750017]; National NSF of China [10531080]; Ministry of Education of China; 973 High Performance Scientific Computation Research Program [2005CB321703]

A high order schema for the numerical solution of the fractional ordinary differential equations

In this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations (FODEs). This technique is based on the so-called block-by-block approach, which is a common method for the integral equations. In our approach, the classical block-by-block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps, while preserving the good stability property of the block-by-block schemes. By using this new approach, we are able to construct a high order schema for FODEs of the order @a,@a>0. The stability and convergence of the schema is rigorously established. We prove that the numerical solution converges to the exact solution with order 3+@a for 0 1. A series of numerical examples are provided to support the theoretical claims.

High-Order Algorithms for Large-Eddy Simulation of Incompressible Flows

Abstract“Defiltering-Transport-Filtering” (DTF) algorithms are proposed for the large eddy simulation of incompressible flows by using high order methods. These new algorithms are based (i) on an approximate deconvolution method for the modeling of the sub-grid scale stress tensor and (ii) on a semi-Lagrangian method to handle the convective term. Such algorithms are implemented in 3D spectral solvers (one homogeneous direction), using differential operators to handle in an approximate way the filtering and defiltering operations. Stability and dissipation properties of the schema are discussed. Preliminary results, obtained with a Chebyshev collocation solver, for the 3D wake of a cylinder with Reynolds number equal to 1000 are presented.

PARALLEL IN TIME ALGORITHM WITH SPECTRAL-SUBDOMAIN ENHANCEMENT FOR VOLTERRA INTEGRAL EQUATIONS

NSF of China [11201077, 11071203, 91130002]; NSF of Fujian Province [2012J01007]; Fuzhou University [0460022456]; Hong Kong Research Grant Council GIF grants; Hong Kong Baptist University FRG grants

Error Analysis of a High Order Method for Time-Fractional Diffusion Equations

In this paper, we consider a numerical method for the time-fractional diffusion equation. The method uses a high order finite difference method to approximate the fractional derivative in time, resulting in a time stepping scheme for the underlying equation. Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials. The main body of this paper is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme. As a by-product and direct extension of our previous work, an error estimate for the spatial discretization is also provided. The key contribution of the paper is the proof of the (

A Laguerre-Legendre Spectral Method for the Stokes Problem in a Semi-Infinite Channel

3-\alpha

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

)-order convergence of the time scheme, where

Spectral methods based on new formulations for coupled Stokes and Darcy equations

\alpha