An Error Estimate of a Eulerian-Lagrangian Localized Adjoint Method for a Space-Fractional Advection Diffusion Equation

Abstract

We derive a Eulerian-Lagrangian localized adjoint method (ELLAM) for a space-fractional advection diffusion equation that includes a fractional Laplacian operator for modeling such application as a superdiffusive advective transport. The method symmetrizes the numerical scheme and generates accurate numerical solutions even if large time steps and relatively coarse grid meshes are used. We also study the structure of the stiffness matrix to further reduce the computational complexity and memory requirement. We prove an error estimate for the ELLAM. Numerical experiments are presented to show the potential of the method.