An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations
This work provides a stable and convergent numerical method for a class of fractional partial differential equations, which is incremental as it combines existing techniques (ADI, spectral, finite difference) for a specific problem type.
The paper develops an alternating direction implicit spectral method for solving two-dimensional multi-term time fractional mixed diffusion and diffusion-wave equations, achieving optimal error of O(N^{-r}+τ^2). Numerical experiments confirm the theoretical analysis.
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 $O(N^{-r}+τ^2)$, where $N, τ, r$ 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.