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.