Combined compact difference scheme for the time fractional convection–diffusion equation with variable coefficients

Abstract

Fourth-order combined compact finite difference scheme is given for solving the time fractional convection–diffusion–reaction equation with variable coefficients. We introduce the flux as a new variable and transform the original equation into a system of two equations. Compact difference is used as a high-order approximation for spatial derivatives of integer order in the coupled partial differential equations. The Caputo fractional derivative is discretized by a high-order approximation. Both Dirichlet and Robin boundary conditions are discussed. Convergence analysis is given for the problem of integer order with constant coefficients under some assumption. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.