A Computational Study of an Implicit Local Discontinuous Galerkin Method for Time-Fractional Diffusion Equations

Leilei Wei,Xindong Zhang

doi:10.1155/2014/898217

Abstract

We propose, analyze, and test a fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional diffusion equation. The proposed method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. By choosing the numerical fluxes carefully, we prove that our scheme is unconditionally stable and convergent. Finally, numerical examples are performed to illustrate the effectiveness and the accuracy of the method.

Highlights

Fractional calculus which is considered as the generalization of the integer order calculus attracts much attention recently its of their numerous applications in physics and engineering
An implicit fully discrete local discontinuous Galerkin (LDG) finite element method is presented for solving a class of time-fractional diffusion equation
Numerical examples show that the combination of the backward differentiation in time and local discontinuous Galerkin (LDG) finite element method in space leads to an approximation of order ((Δt)2−α + hk+1) for smooth enough solution

Summary

Introduction

Fractional calculus which is considered as the generalization of the integer order calculus attracts much attention recently its of their numerous applications in physics and engineering They provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. Machado et al [20] introduced the recent history of fractional calculus; as for the detailed theory and applications of fractional integrals and derivatives, we can refer to [21, 22] and the references therein Due to their numerous applications in the areas of physics and engineering, solving such equations and numerical schemes for fractional differential equations has been stimulated. We propose a fully discrete local discontinuous Galerkin (LDG) finite element method for solving the time-fractional diffusion equation.

Notations and Auxiliary Results

Fully Discrete LDG Scheme

Numerical Examples

Conclusion