An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations

In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is $O(N^{-r}+{\tau }^2)$, where $N,\tau ,r$ are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids.