New High-Order Compact ADI Algorithms for 3D Nonlinear Time-Fractional Convection-Diffusion Equation

Shuying Zhai,Zhifeng Weng,Xinlong Feng

doi:10.1155/2013/246025

Shuying Zhai, Zhifeng Weng + Show 1 more

Open Access

https://doi.org/10.1155/2013/246025

Copy DOI

Abstract

Numerical approximations of the three-dimensional (3D) nonlinear time-fractional convection-diffusion equation is studied, which is firstly transformed to a time-fractional diffusion equation and then is solved by linearization method combined with alternating direction implicit (ADI) method. By using fourth-order Padé approximation for spatial derivatives and classical backward differentiation method for time derivative, two new high-order compact ADI algorithms with ordersO(τmin(1+α,2−α)+h4)andO(τ2−α+h4)are presented. The resulting schemes in each ADI solution step corresponding to a tridiagonal matrix equation can be solved by the Thomas algorithm which makes the computation cost effective. Numerical experiments are shown to demonstrate the high accuracy and robustness of two new schemes.

Highlights

In this paper, we consider numerical approximations of the following 3D nonlinear time-fractional convection-diffusion equation: ∂αu ∂tα 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨Ω×(0,T] (1)= (Δu − pux − quy − ruz + su + φ (u) + f)󵄨󵄨󵄨󵄨󵄨Ω×(0,T], with the initial and boundary conditions u (x, y, z, 0)󵄨󵄨󵄨󵄨Ω = u0 (x, y, z)󵄨󵄨󵄨󵄨Ω, u (x, y, z, t)󵄨󵄨󵄨󵄨∂Ω×(0,T] = 0, (2)where 0 < α < 1, p, q, r, and s are constants, and φ(u) is some reasonable nonlinear function of u = u(x, y, z, t)
We focus on the 3D nonlinear timefractional convection-diffusion equation
We propose a transformation based on the compact finite difference method [35]

Summary

Introduction

We consider numerical approximations of the following 3D nonlinear time-fractional convection-diffusion equation:. For the two-dimensional (2D) FDEs, Zhuang and Liu [19] considered a time-fractional diffusion equation on a finite domain, and a first-order implicit difference approximation was proposed to solve the equation. Cui [27] proposed a high-order ADI compact finite difference scheme with order O(τ2−α + h4) for time-fractional diffusion equation. The main purpose of this paper is to solve the nonlinear time-fractional convection-diffusion problem using the Padeapproximation combined with ADI method and the onedimensional tridiagonal Thomas algorithm. We propose a transformation based on the compact finite difference method [35] In this method, the original time-fractional convection-diffusion equation is transformed to a time-fractional diffusion equation; the resulting time-fractional diffusion problem is solved by an efficient high-order method.

Two New High-Order Compact ADI Algorithms

Numerical Experiments

Conclusions