Discontinuous Galerkin method for fractional convection-diffusion equations

2013/04/22 by Qiuyan Xu, Jan S. Hesthaven, Xu, Q. +1
Engineering · Mathematics · #26A33 #35R11 #65M12 #65M60 #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1304.6047

openalex publication_date 2013/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We propose a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order α(1<α<2) defined through the fractional Laplacian. The fractional operator of order α is expressed as a composite of first order derivatives and fractional integrals of order 2-α, and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin method scheme is derived for the equations. We prove stability and optimal order of convergence O(h^k+1) for subdiffusion, and an order of convergence of \cal O(h^k+1/2) is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.

Citations

• Error Estimates to Smooth Solutions of Runge--Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws
• Finite Element Method for the Space and Time Fractional Fokker–Planck Equation

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