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: partial derivative(alpha)(t)u - div(A del u) = f, 0 < alpha < 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 LI difference approximation to partial derivative(alpha)(t)u. We first establish a new error bound of 0 (Delta t(1+delta-alpha))-order for the Ll formula under the condition of u(t) epsilon C-1,C-8[0,T] where delta epsilon [0, 1] is the Holder 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 0 (Delta t(1+delta-alpha) + h(2) )-order in the L2 -norm. Numerical examples are provided to support our theoretical analysis. (C) 2018 Elsevier Inc. All rights reserved.