Summary: 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 \(\beta \in [1, 2]\). 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 \(\mathcal O(h^{k + 1})\) uniformly across the continuous range between pure advection \(({\beta} = 1)\) and pure diffusion \(({\beta} = 2)\). 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.