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

•Combined compact finite difference scheme for solving the time fractional convection–diffusion–reaction equation is given.•The primary unknown and its derivative are solved simultaneously.•The coefficients of the fractional partial differential equation are variable.•Both Dirichlet and Robin boundary conditions are discussed, and the algorithm gives high-order convergence results.

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.