A high-order nodal discontinuous Galerkin method for a linearized fractional Cahn-Hilliard equation
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. Received August 13, 2012. Revised May 21, 2013. Published online October 7, 2013.