Finite difference schemes for the two-dimensional multi-term time-fractional diffusion equations with variable coefficients

Two implicit finite difference schemes for solving the two-dimensional multi-term time-fractional diffusion equation with variable coefficients are considered in this paper. The orders of the Riemann–Liouville fractional time derivatives acting on the spatial derivatives can be different in various spatial directions. By integrating the original partial differential equation with time variable first, and the second-order spatial derivatives are approximated by the central difference quotients, then the fully discrete finite difference scheme can be obtained after the right rectangular quadrature formulae are used to approximate the resulting time integrals. The convergence analysis is given by the energy method, showing that the difference scheme is first-order accurate in time and second order in space. Based on a second-order approximation of the Riemann–Liouville fractional derivatives using the weighted and shifted Grünwald difference operator, we present the Crank–Nicolson scheme and prove it is second-order accurate both in time and space. Numerical results are provided to verify the accuracy and efficiency of the two proposed algorithms. Numerical schemes and theoretical analysis can be generalized for the three-dimensional problems.