Speaker: Dr. Qinwu Xu (School of Mathematical Science, PKU)

Time: 14:00--15:00, May 26

Venue: Room 29 at Quan Zhai, BICMR

Abstract: In this talk, a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials. Based on the proposed method, a high order discretization is obtained for fractional Laplacian. Then, a discontinuous Galerkin method is proposed for fractional convection-diffusion equations with a superdiffusion operator of order α (1 < α < 2) defined through the fractional Laplacian. The fractional Laplacian of order α is expressed as a composite of first order derivatives and a fractional integral of order 2 − α. The fractional convection-diffusion problem is expressed as a system of low order differential/integral equations, and a local discontinuous Galerkin method is proposed for the equations. We prove stability and optimal order of convergence O(h^(k+1)) for the fractional diffusion problem, and an order of convergence of O(h^(k+1/2)) is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.