Original articles
Local discontinuous Galerkin method for multi-term variable-order time fractional diffusion equation

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Abstract

This paper presents an effective numerical method for multi-term variable-order time fractional diffusion equations with the variable-order fractional derivative. The local discontinuous Galerkin method and the finite difference method are used in the spatial and temporal directions, respectively. We prove that the scheme is unconditional stable and convergent with O(hs+1+(Δt)2r), where r=max{ɛ(t)}. s, h, Δt are the degree of piecewise polynomials, the space step sizes, and the time step sizes, respectively. Some numerical experiments are used to illustrate the effectiveness and applicability of the scheme.

Introduction

Fractional calculus which involves derivatives and integrals of any order can be applied in many fields to describe different problems. It has received extensive attention from various disciplines, such as physics, engineering science, and mathematics. It was produced due to discussions between L’Hopital and Leibniz, and was promoted by famous scientists such as Euler, Abel, and Laplace [29]. In recent years, many scholars have also paid attention to fractional calculus and used it to describe different types of problems. Fang [6] reduced the number of required parameters by using the fractional calculus to the viscoelastic constitutive model. Chen [4] applied fractional calculus to the supply chain financial system and established a three-dimensional supply chain game model which is closer to reality. Lu et al. [19] analyzed complex biological systems and modeled artificial neural networks by it. There are many other applications, such as diffusion in porous media [20], pollutant transport kinetics [36], and so on.

Samko [23] studied the fractional integration and differentiation of variable-order, and discussed its mathematical analysis. The later research results have shown that many complex physical phenomena can be better described by using variable-order fractional differential equations, such as signature verification, algebraic structure, viscoelastic materials, and noise reduction [7], [9], [24], [41]. Variable order fractional derivative which changes with spatial and temporal variables is a good tool to eliminate the nonphysical singularity of the solutions for constant-order FPDEs and model multi-physics phenomena, and has many advantages in characterizing memory property of systems [25], [26], [27], [30].

Numerical methods play an important role in studying partial differential equations with variable order derivatives. Langlands and Henry proposed a finite-difference scheme for fractional-order diffusion equations and studied the accuracy and stability of the scheme [11]. Cao and his collaborators studied a compact finite-difference scheme for variable-order diffusion equations [1]. Li et al. solved the nonlinear time fractional parabolic problem using the L1-discontinuous finite element method and obtained the optimal error estimate [13]. Du et al. [5] studied the implicit scheme and Crank–Nicolson scheme for the variable-order time fractional diffusion equation over the finite domain. Haq et al. [10] used finite difference method to solve the variable order time fraction (1+1)-dimensional and (1+2)-dimensional convection–diffusion equations. Chen and Liu [3] proposed a fully discrete dual-grid modified characteristic curve method based on two-dimensional nonlinear variable-order time fractional convection–diffusion equations. Sadri and Aminikhah [22] studied an algorithm based on Chebyshev polynomials for varied-order time fractional diffusion wave equations. Chen [2] solved the variable-order anomalous subdiffusion equation with first order temporal accuracy and fourth order spatial accuracy. Lin and Liu [16] studied the explicit finite difference method for variable-order nonlinear fractional differential equations. Wang and Zheng [30] proved the wellposedness of a nonlinear variable-order fractional differential equation and proposed a finite difference scheme for the problem.

Although there are some numerical algorithms for variable-order partial differential equations, it is still challenging to construct high-order numerical schemes for the models. The discontinuous Galerkin method is a hybrid of the finite element and finite volume methods, with many of the benefits of each. The resulting equations are local to the produced elements, which is a significant distinction between the DG method and the traditional finite element method. Without looking at adjoining elements, the solution can be reconstructed in each element. The discontinuous Galerkin method is a very attractive tool for solving partial differential equations [38] and has been proposed to numerically simulate fractional differential equations [8], [14], [15], [17], [18], [31], [34] due to its flexibility and efficiency in terms of meshes and shape functions. Readers may refer to [12], [21], [28], [32], [35], [37], [40] for additional information.

In this paper, we will construct a high-order local discontinuous Galerkin method for the following multi-term variable-order time-fractional diffusion equations 0CDtɛ(t)u(x,t)+i=1Iai(t)0CDtɛi(t)u(x,t)2u(x,t)x2=g(x,t),(x,t)(a,b)×(0,T],u(x,0)=u0(x),x[a,b],where 0<ɛ1(t)ɛI(t)ɛ(t)<1 are the orders of the fractional derivatives, ai(t) are continuous functions of t in the interval [0,T], r=max{ɛ(t)}, and g,u0 are smooth functions. The solution in this paper is considered to be either periodic or compactly supported.

The variable-order Caputo fractional derivative in (1.1) is defined by 0CDtη(t)u(x,t)=1Γ(1η(t))0tu(x,ξ)ξdξ(tξ)η(t).

A brief outline of the paper is as follows. In Section 2, some symbols and auxiliary results are described. Then in Section 3, we present the fully discrete local discontinuous Galerkin method for the multi-term variable-order time fractional diffusion equation. In Section 4, we proved that the scheme is unconditionally stable and convergent. Some numerical examples are given to demonstrate the reliability and efficiency of the method in Section 5, and the last section includes concluding comments.

Section snippets

Notations and auxiliary results

Let a=x12<x32<<xN+12=b be a partition of Ω=[a,b], denote Ij=[xj12,xj+12], for j=1,,N, and Δxj=xj+12xj12,1jN, h=max1jNΔxj.

Denote by uj+12+ and uj+12 the values of u at xj+12, from the right cell Ij+1, and the left cell Ij, respectively.

The associated discontinuous Galerkin space Vhs: Vhs={v:vPs(Ij),xIj,j=1,2,,N}.

Regarding the error estimation, we will use two projections in the one-dimensional interval [a,b], denoted by P:Hs+1(D)Vhs, which means that for each j Ij(Pϖ(x)ϖ(x))v(x)=

The schemes

Next, we introduce the numerical scheme of (1.1) solution.

We divide the interval [0,T] evenly into time step size Δt=TM,MN, tn=nΔt,0nM,nN be the mesh points.

For 0<η(t)<1, we have 0CDtη(tn)u(x,tn)=1Γ(1η(tn))0tnu(x,ξ)ξ(tnξ)η(tn)dξ=1Γ(1η(tn))s=0n1tsts+1u(x,ξ)ξ(tnξ)η(tn)dξ=1Γ(1η(tn))s=0n1tsts+1u(x,ts+1)u(x,ts)Δt(tnξ)η(tn)dξ+rηn=1Γ(1η(tn))1Δts=0n1(u(x,ts+1)u(x,ts))1η(tn)1((tnts+1)1η(tn)(tnts)1η(tn))+rηn=Δtη(tn)Γ(2η(tn))s=1n(u(x,ts)u(x,ts1))((ns+1)1η(tn)(ns)

Stability and convergence

In order to simplify the notation we consider the case of g=0 in the numerical analysis.

Theorem 4.1

Under periodic or compactly supported boundary conditions, the fully discrete locally discontinuous Galerkin scheme (3.4) is unconditionally stable, and the numerical solution uhn satisfies uhnuh0,n=1,2,,M.

Proof

Take the test functions v=uhn,ξ=β1zhn in the scheme (3.4), and use the fluxes (3.5), we get β0uhn2+β1zhn2+β1ΨΩ(uhn,zhn;zhn,uhn)s=1n1(ωns1ɛ(tn)ωnsɛ(tn)+i=1IΓ(2ɛ(tn))Γ(2ɛi(tn))ai(tn)(ωns

Numerical experiment

In this section, some numerical examples are presented to illustrate the accuracy and performance of the method. From the results of different numerical examples, we can see that the scheme is convergent and effective.

Example 5.1

Consider the problem (1.1) 0RDtɛ(t)u(x,t)+i=1Iai(t)0RDtɛi(t)u(x,t)uxx=g(x,t),(x,t)(0,1)×(0,1].In the interval x(0,1), the initial condition u(x,0)=0. Let the function g(x,t)=2x2(1x)2(t2ɛ(t)Γ(3ɛ(t))+a1(t)t2ɛ1(t)Γ(3ɛ1(t)))t2(12x212x+2),then the exact solution is u(x,t)=t2x

Conclusion

In this paper, we propose a high order numerical method to solve a class of multi-term variable-order fractional diffusion equations. By using finite difference method in time and local discontinuous Galerkin method in space, we get a fully discrete LDG scheme. By taking the appropriate numerical flux, we prove that the method is stable and convergent with O(hs+1+(Δt)2r). Some numerical experiments are also displayed which is consistent with the theoretical results.

Acknowledgments

This work is supported by the Training Plan of Young Backbone Teachers in Henan University of Technology, PR China (21420049), the Training Plan of Young Backbone Teachers in Colleges and Universities of Henan Province, PR China (2019GGJS094), Scientific and Technological Research Projects in Henan Province (212102210612) and the National Natural Science Foundation of China (12126325, 12126315).

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