Efficient compact finite difference method for variable coefficient fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions in conservative form

Abstract

In this paper, a high-order compact finite difference scheme is derived for the time fractional sub-diffusion equation with nonhomogeneous Neumann boundary conditions in conservative form. Based on the L2- $$1_{\sigma }$$ approximation formula of the time fractional derivative, a high-order efficient compact finite difference method is developed. The unconditional stability of the resulting scheme and its convergence of second order in time and fourth order in space are rigourously proved by a discrete energy analysis method. The theoretical results are illustrated by examples.