The finite difference/finite volume method for solving the fractional diffusion equation

In this paper, we study the finite volume method for solving the time-fractional diffusion equation: ∂tαu−div(A∇u)=f, 0<α<1. We present and analyze a fully discrete numerical scheme which is based on the linear finite volume method for the spatial discretization and the L1 difference approximation to ∂tαu. We first establish a new error bound of O(△t1+δ−α)-order for the L1 formula under the condition of u(t)∈C1,δ[0,T] where δ∈[0,1] is the Hölder continuity index. Then, we prove that this fully discrete finite volume scheme is unconditionally stable and the discrete solution admits the optimal error estimate of O(△t1+δ−α+h2)-order in the L2-norm. Numerical examples are provided to support our theoretical analysis.