High-order compact finite difference method for the multi-term time fractional mixed diffusion and diffusion-wave equation

In this paper, the multi-term time fractional mixed diffusion and diffusion-wave equation is investigated. Firstly, a compact finite difference scheme with fourth-order spatial accuracy and high-order temporal accuracy is derived. Then, the unconditional stability and convergence in the maximum norm of the derived high-order compact finite difference method have been discussed rigorously by means of the energy method. Numerical experiments are presented to test the convergence order in the temporal and spatial direction, respectively. Additionally, to precisely demonstrate the computational efficiency of the derived high-order compact finite difference method, the maximum norm error and the CPU time are measured in contrast with the second-order finite difference scheme for the same temporal grid size. Finally, a practical example is presented to show the applicability of the time fractional mixed diffusion and diffusion-wave model and the efficiency of the derived high-order compact finite difference method.