1 School of Mathematics and Statistics, Lanzhou University,
Lanzhou 730000, People’s Republic of China; Division of Applied Mathematics, Brown
University, 182 George Street, Providence, RI 02912, USA.
2 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.
Revised: 21 May 2013
We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems in one space dimension, characterized by having fractional derivatives, parameterized by β ∈[1, 2]. After demonstrating that a classic approach fails to deliver optimal order of convergence, we introduce a modified local numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pure advection (β = 1) and pure diffusion (β = 2). In the two classic limits, known schemes are recovered. We discuss stability and present an error analysis for the space semi-discretized scheme, which is supported through a few examples.
Mathematics Subject Classification: 35R11 / 65M60 / 65M12
Key words: Fractional derivatives / local discontinuous Galerkin methods / stability / convergence / error estimates
© EDP Sciences, SMAI 2013