An alternating direction implicit compact finite difference scheme for the multi-term time-fractional mixed diffusion and diffusion-wave equation

An alternating direction implicit(ADI) compact finite difference scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion-wave equation is given in this paper. By using the Riemann-Liouville fractional integral operator on both sides of the original equation, we obtain a time-fractional integro-differential equation with the order of the highest derivative being one. Using the weighted and shifted Grunwald formulas of Riemann-Liouville fractional derivative and fractional integral, and the Crank-Nicolson approximation, a temporal second-order approximation is obtained for the equivalent integrodifferential system. In the spatial direction, a fourth-order compact approximation is employed to give a fully discrete scheme. Using the splitting techniques for higher dimensional problems, we derive the fully discrete ADI Crank-Nicolson scheme. With the positive definiteness of the related time discrete coefficients and spatial operators, the stability and convergence (second-order in time and fourth-order in space) in the discrete L2-norm are proved by using energy method. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed method. & COPY; 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.