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

TL;DR

This paper introduces an ADI spectral method for efficiently solving complex two-dimensional multi-term time fractional equations, with proven stability, convergence, and applicability to modeling viscoelastic fluid transport.

Contribution

It develops a novel ADI spectral scheme combining Legendre spectral approximation and finite difference time discretization for multi-term fractional equations.

Findings

Proves stability and convergence of the scheme.

Achieves optimal error of O(N^{-r} + τ^2).

Validates the method with numerical experiments.

Abstract

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.