Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

Gang Dong,Zhichang Guo,Wenjuan Yao

doi:10.1515/math-2021-0036

Abstract

Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

Highlights

Fractional differential equations have attracted considerable interest due to their ability to model many phenomena
Numerical methods for TFCD with high-order accuracy 783 of approximation in space and second order in time were constructed by Alikhanov in [14], where the stability and convergence were studied by the method of energy inequalities
We proved that the alternating direction implicit (ADI) scheme is unconditionally stable to the initial values and the inhomogeneous term

Summary

Introduction

Fractional differential equations have attracted considerable interest due to their ability to model many phenomena They are widely applied in various fields of science and engineering, such as signal processing, anomalous diffusion, wave propagation, and turbulence [1,2,3]. These equations are of three types that contain derivatives of fractional order in space, time, or space-time [4]. Because of the nonlocal nature of fractional differential operators, analytical solutions of these equations are not available in most cases. A number of authors proposed numerical methods for solving fractional diffusion equations [5,6,7,8,9,10,11]

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