Elsevier

Applied Mathematics and Computation

Volume 335, 15 October 2018, Pages 305-312
Applied Mathematics and Computation

An approximation scheme for the time fractional convection–diffusion equation

https://doi.org/10.1016/j.amc.2018.04.019Get rights and content

Abstract

In this paper, a discrete form is proposed for solving time fractional convection–diffusion equation. Firstly, we obtain a time discrete scheme based on finite difference method. Secondly, we prove that the time discrete scheme is unconditionally stable, and the numerical solution converges to the exact one with order O(τ2α), where τ is the time step size. Finally, two numerical examples are proposed respectively, to verify the order of convergence.

Introduction

Many natural phenomena are modeled by partial differential equations, even medical problems [1], [2] can also be modeled by partial differential equations. As a class of partial differential equations, integer order convection–diffusion equations are also have a wide range of applications. The growing number of applications of fractional derivatives in various fields [3], [4], [5], [6] such that fractional convection–diffusion equations have received an increasing attention in recent years and have been used to model a wide range of problems in sound, heat, electrostatics, electrodynamics, fluid flow, and elasticity.

One of the key issues with fractional convection–diffusion equations is the design of efficient numerical schemes and computational efficiency is an important parameter for the numerical schemes. A variety of numerical methods for the fractional partial differential equations have been proposed, such as finite element methods [7], [8], [9], [10], [11], [12], finite difference methods [13], [14], [15], [16], [17], spectral methods [18], [19], [20], meshless methods [21], [22]. In [13], fractional advection–diffusion equation with space–time fractional derivative has been considered and the stability and convergence of the mentioned method has also been discussed. Zhang et al. [14] have proposed a new numerical method for the time mobile–immobile advection–dispersion problem with variable fractional derivative. Zhuang et al. [15] have developed some numerical methods for the nonlinear fractional advection–diffusion equation with variable-order fractional derivative. The authors of [23] have proposed an implicit meshless method based on the radial basis functions for the numerical simulation of time fractional diffusion equation.

In this paper, we aim to propose a time discrete form combined with finite difference methods for the time fractional convection–diffusion equation. We consider the time fractional convection–diffusion equation of the form: {αu(x,t)tα=Δu(x,t)u(x,t)+f(x,t),(x,t)Ω×[0,T],u(x,0)=u0(x),xΩ,u(x,t)Ω=0,t[0,T], where Ω denotes a bounded domain in Rn, 0 < α < 1, f and u0(x) ≡ u0 are given smooth functions, Δu(x,t)=2u(x,t)x12++2u(x,t)xn2 and u(x,t)=u(x,t)x1++u(x,t)xn. The fractional derivative αu(x,t)tα is Caputo fractional derivative defined by αu(x,t)tα=1Γ(1α)0tu(x,s)sds(ts)α.

Section snippets

Time discrete scheme

We will introduce a finite difference approximation to discrete the time fractional derivative. Let tm ≔ , where τ ≔ T/M is the time step, m=0,1,,M be mesh points, where M is a positive integer.

The time fractional derivative αutα at tm+1 is estimated by αu(x,tm+1)tα=1Γ(1α)k=0mtktk+1u(x,s)sds(tm+1s)α=1Γ(1α)k=0mu(x,tk+1)u(x,tk)τtktk+1ds(tm+1s)α+Rτ(1).Using the analysis in [18] the truncation error Rτ(1) has the following form |Rτ(1)|C0|1Γ(1α)k=0mtktk+12stk+1tk(tm+1s)αds+O(

Stability analysis and error estimates

In this section, we discuss the stability and convergence of Eq. (6) using the following lemma. Firstly, we define the functional spaces endowed with standard norms and inner products. The space L2(Ω) is equipped with the usual L2-scalar product (u,v)=ΩuvdΩ,which induces the L2-norm u2=(u,u)1/2,for u, v ∈ L2(Ω).

Now by simplifying Eq. (6), we have um+1λΔum+1+λum+1=(1b1)um+k=1m1(bkbk+1)umk+bmu0+λfm+1,m=1,2,,M.

Lemma 3.1

[24] Let Ω be a bounded domain in Rn with piecewise smooth boundary ∂Ω, if U

Numerical results

In this section, we carry out two numerical experiments and present some results to confirm our theoretical statements. In our examples, we use the method of the order of time equal to space order [13], the main purpose is to check the convergence behavior of the numerical solution with respect to the time step τ used in the calculation.

Conclusion

In this paper, we proposed a discrete scheme to solve the fractional convection–diffusion equation. The stability analysis and error estimates for the discrete scheme are discussed. Numerical examples for some case are given to illustrate the accuracy and capability of the method. Constructing more efficient algorithms to reduce the storage requirement is also our goal in future works.

Acknowledgments

Authors thank the anonymous referees and editors very much for their valuable suggestions and comments, which greatly helped us to improve this work. This work is supported by the NSFC (No.11461072), the Youth Science and Technology Education Project of Xinjiang (No.QN2016JQ0367), the Science and Technology Innovation Project for Master Graduate Student of Xinjiang Normal University (No. XSY201602011).

References (24)

Cited by (0)

This work is supported by the NSFC (No. 11461072), the Youth Science and Technology Education Project of Xinjiang (No. QN2016JQ0367) and the Science and Technology Innovation Project for Master Graduate Student of Xinjiang Normal University (No. XSY201602011).

View full text