ERROR ANALYSIS OF A HIGH ORDER METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS

作者：Lv, Chunwan; Xu, Chuanju^*

来源：SIAM Journal on Scientific Computing, 2016, 38(5): A2699-A2724.

DOI：10.1137/15M102664X

• time-fractional diffusion equation
• time stepping scheme
• spectral methods
• error estimates

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 (3 - alpha)-order convergence of the time scheme, where a is the order of the time-fractional derivative. Then the theoretical result is validated by a number of numerical tests. To the best of our knowledge, this is the first proof for the stability of the (3 - alpha)-order scheme for the time-fractional diffusion equation.

• 出版日期2016
• 单位厦门大学