The Upwind Finite Volume Element Method for Two-Dimensional Time Fractional Coupled Burgers’ Equation

The ﬁnite volume element method for approximating a two-dimensional time fractional coupled Burgers’ equation is presented. The linear ﬁnite volume element method is used for spatial discretization and the upwind technique is used for the nonlinear convective term to get the semi-discrete scheme. Further, the time-fractional derivative term is approximated by using L1 formula and the nonlinear convection term is treated by linearized upwind technique to get the fully discrete scheme. We prove that the semi-discrete scheme is convergent with one-order accuracy in space and the fully discrete scheme is convergent with one-order accuracy both in time and space in L 2 -norm. Numerical experiments are presented ﬁnally to validate the theoretical analysis.


Introduction
In this paper, we consider the following two-dimensional time fractional coupled Burgers' equation: where Ω ⊂ R 2 is a bounded domain with boundary ∂Ω, ν > 0 is the viscosity.Symbols ∆, ∇ denote the Laplacian and gradient operators, respectively.u = u(x, t) = (u(x, t), v(x, t)) is the unknown vector function which represents the velocity of the fluid.The Caputo fractional derivative C 0 D α t is defined as Fractional differential operators, different from integer differential operators, are non-local and very suitable for describing materials with memory and heredity in the real world.Fractional differential equations are widely used in the fields of anomalous diffusion, viscoelastic mechanics, fluid mechanics, boundary layer effect of pipeline, signal recognition processing and system identification (Baleanu et al., 2012;DElia, 2020;Li, Changpin and Chen, An, 2018) etc. Burgers' equation is the simplest nonlinear time dependent partial differential equation which was first raised by Bateman in (Bateman, Harry, 1915) when he mentioned it as worthy of study and gave a special solution.It exists in plenty areas of applied mathematics, such as acoustic waves, heat conduction and modelling of dynamics (Caldwell, J et al. 1981;Cole, Julian D, 1951).Fractional Burgers' equation can be used to describe the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe.The fractional derivative results from the cumulative effect of the wall friction through the boundary layer (Sugimoto, Nobumasa, 1991;Inc, Mustafa, 2008).Therefore, fractional Burgers' equations have been taken more and more seriously attention and numerical methods have been developed to provide numerical solutions for them.For example, Wang (Wang, Qi, 2006) discussed the adomian decomposition method for time and space fractional Burgers' equations.Esen and Tasbozan (Esen et al., 2015) proposed quadratic B-spline Galerkin method for one-dimensional time fractional Burgers' equation.Li et al. (Li, Dongfang et al. 2016) employed a linear implicit finite difference scheme for solving the generalized time fractional Burgers' equation.Cao et al. (Cao, Wen et al., 2017) presented discontinuous Galerkin method to solve a two-dimensional time fractional Burgers' equation with high and low Reynolds numbers on quasi-uniform triangular mesh.Wang et al. (Wang, Haifeng et al., 2021) proposed the weak Galerkin finite element method for a class of one-dimensional time fractional generalized Burgers' equation.Hussein and Ahmed Jabbar (Hussein, Ahmed Jabbar, 2020) proposed a weak Galerkin finite element method for solving two-dimensional time fractional coupled Burgers' equations on uniform triangular mesh.Qiao and Tang (Qiao, Leijie et al., 2022) introduce an accurate, robust, and efficient finite difference scheme with graded meshes for the time-fractional Burgers' equation.Zhang and Feng (Zhang, Yadong et al., 2023) present a local projection stabilization virtual element method for the time-fractional Burgers' equation with high Reynolds numbers.Kashif and Dwivedi (Mohd, Kashif etal., 2022) use non-standard finite difference scheme and Fibonacci collocation method for fractional order Burgers equation.AA AL-saedi and J Rashidinia (AA, AL-saedi etal., 2023) present a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burgers equation.
The finite volume element method (FVEM) (Li, Ronghua et al., 2000;Cai, Zhiqiang et al., 1991;Cai, Zhiqiang, 1990;Ewing, Richard et al. 2000) is an important numerical method for solving partial differential equations.In some literature, FVEM is also called the control volume method, the covolume method, and the first-order generalized difference method.FVEM has been widely applied due to its various important features.For example, the grid is flexible and it inherits some physical conservation laws of the original problems locally which are very preferable in practical applications.And the computational effort is greater than in finite difference methods and less than finite element methods, while the accuracy is higher than finite difference methods and nearly the same as finite element methods.Liang (Liang, D, 1990;Liang, D, 1991) combined the upwind technique and the FVEM to solve the linear convection-dominated problems.Yang (Yang, Qing, 2013) used the upwind finite volume element method for two-dimensional integer-order Burgers' equation.The nonlinear convective term was dealt with upwind technique in order to avoid numerical oscillation.For the study of problem (1.1)-(1.3),there are still gaps in finite volume element method.In this paper, we will consider upwind finite volume element method for the approximation of the two-dimensional time coupled fractional Burgers' equation (1.1).The upwind approximation is applied to handle the nonlinear convection term in view of its advantage to solve the convection-dominated problems.The semi-discrete and fully discrete schemes are defined.The linear finite volume element format is used for spatial discretization, and the upwind technique is used for the nonlinear convective term to get the semi-discrete scheme.Then L1 formula is used to approximate the time-fractional derivative term, and the nonlinear convection term is treated by linearized upwind technique to get the fully discrete scheme.We prove that the semi-discrete scheme is convergent with one-order accuracy in space in L 2 -norm and the fully discrete scheme is convergent with oneorder accuracy both in time and space in L 2 -norm.Some numerical experiments show that our method is effective for time fractional Burgers' equation.
The outline of this paper is as follows.In Section 2, we build the upwind finite volume element semi-discrete scheme and fully discrete scheme for the coupled fractional Burgers' equation, respectively.In section 3, we derive the L 2 norm error estimates for the semi-discrete scheme and the fully discrete scheme, respectively.In section 4, numerical experiments are presented to show the efficiency of proposed method and confirm our theoretical analysis.

The Approximation Schemes
In this section, we introduce some important function spaces and give the semi-discrete scheme and the fully discrete scheme for problem (1.1)-(1.3).
is the set of all nodal points of T h , where h = max h K , h K is the diameter of element K. Associated with the triangulation T h , we introduce the velocity approximation space: where P 1 (K) is the set of linear polynomials on element K.
In order to define the finite volume element method, we need a dual partition associated with the primal partition T h .We construct the barycenter dual partition T * h by connecting the barycenter to the midpoints of edges of each K ∈ T h by straight lines.Thus, for each nodal point P in T h , there exists a polygonal surrounding it, which is called the dual element or the control volume at point P and denoted by K * P .So we have T * h = {K * P , P ∈ N h }.The test function space is defined as where P 0 (K * P ) is the constant set on element K * P .Clearly, the dimensions of X h and V h are the same.Introduce the interpolation operators R h : (C( Ω)) 2 → X h and γ: X h → V h , respectively.Assuming that u ∈ (H 2 (Ω)) 2 , we can easily get the following interpolation estimates: (2.1) Lemma 1. (Li, Ronghua et al., 2000) (i) The mapping γ is self-adjoint with respect to the L 2 -inner product in X h : where the positive constant C 1 and C 2 are independent of h.
We start by testing the Eq.
Using integration by parts and the fact that γv is a piecewise constant function on T * h , we get where < ∇u • n, γv >= So (2.5) can be rewritten as Lemma 2. (Li, Ronghua et al., 2000) For the bilinear form A(•, •), we have the following conclusions.

8)
(iii) There exists a positive constant µ such that (2.9) (2.10) Proof.It is known by the definition of the fractional derivative that We will approximate b 2 (u, v, w) by using the upwind technique.Let Λ i = j : P j is adjoint with P i }.Assuming that j ∈ Λ i , let Γ i j = ∂K * P i ∩ ∂K * P j and γ i j is the length of Γ i j .Denote by n i j the unit outward normal vector of Γ i j when Γ i j is regarded as the boundary of K * P i .Define The upwind approximation form of the nonlinear term b 2 (u, v, w) is defined by the form Using the Heaviside function (2.12) Then, the semi-discrete finite volume element scheme of problem (1.1) (2.14) Let T τ = {t n |t n = nτ, 0 ≤ n ≤ N} be a uniform partition of [0, T ] with the time step τ = T/N.Based on a piecewise linear interpolation, the L1-approximation to the Caputo fractional derivative is given by ), the truncation error Q n satisfies (Sayevand, 2016) For a sequence {ω n } N n=0 , we define where With the above notations, the fully discrete finite volume element scheme for (1.1)−(1.3)seeks u n h ∈ X h such that (2.17)

Error Analysis
In this section, we present the error estimates for the semi-discrete scheme (2.13)-(2.14)and the fully discrete scheme (2.16)-(2.17) in L 2 -norm.

Error Estimates for the Fully Discrete Scheme
Lemma 7. (Li, Dongfang, 2018) Suppose that the nonnegative sequence {ω n } N n=0 and {g n } N n=0 satisfy where λ 1 and λ 2 are both positive constants independent of the time step τ.Then, there exists a positive constant τ * such that, when τ ≤ τ * , Theorem 8. Let u and {u n h } N n=0 be the solutions of (1.1)-( 1.3) and (2.16)-(2.17),respectively.Also assume that u satisfies the necessary regularities and the discretization parameters obey the relation τ = O(h).Then where C depends on Proof.Subtract (2.16) from (2.6) to obtain that (3.31) Taking v h = e n , we have (3.32) For the left two terms of (4.32), from Lemma 3, we have and The five terms on the right-hand side of (3.32) are represented by T 1 , T 2 , ..., T 5 , respectively.So, (3.32) can be rewritten as Now, let's estimate the terms T 1 , T 2 , ..., T 5 one by one.Using the property of L1-approximation we have For T 2 , we first have (3.37) Then using Young's inequality we get (3.38) For the next term, by applying Lemma 2, we have Next, we make the following induction hypothesis: For T 4 , using the similar argument as (3.15) and noting (3.40), we deduce that (3.41) Next, we write where n , e n .E 1 and E 2 can be handled as D 1 and D 2 in Theorem 6.Thus, we have and ≤ C e n 2 + 2ε e n 2 1 . ( (3.45) Taking ε = µ/6, we have By Lemmas 1 and 7, there exists a positive constant τ * such that, when τ < τ * , e n ≤ C(τ + h). (3.47) Now we prove the introduction hypothesis (3.40).Noting that u 0 h = R h u 0 , so it holds true when s = 0 obviously.Suppose it holds when 0 ≤ s ≤ n − 1, from (3.47) and the assumption τ = O(h), we have So it is true when 0 ≤ s ≤ n.Thus, we know that the hypothesis (3.40) holds for any 1 ≤ n ≤ N. Using triangular inequality and the interpolation theory we get (3.49)

Numerical Examples
In this section, we will show the numerical performances of proposed finite volume element method on the convergence and efficiency by several examples.
Table 1.Numerical results for α = 0.3 and α = 0.7 Example 2. We consider the following time fractional Burgers' equation: where Ω = (0, 1) × (0, 1), T = 1 and ν = 0.01.In this example, the exact solution is not given beforehand, so we take the numerical solution with h = 1 150 as the reference solution to calculate the error and convergence rate.In Table 2, we present the numerical results for α = 0.2 and α = 0.8.It can be seen in the table that the format is convergent with one-order accuracy.Surface plots of the reference solution u and numerical solution u h for α = 0.8 are given in Figure 4 and Figure 5.As can be seen from the table and figure, it is consistent with the results of our theoretical analysis.Table 2. Numerical results for α = 0.2 and α = 0.8 In order to show that our method keeps stable when ν is smaller, we give the the numerical solution by FVEM with and without upwinding technique for ν = 0.001 in Figure 6 and Figure 7.We can see that our method can perform well but the approximation without upwinding technique produces nonphysical oscillations.These are seen much more clearly in Figure 8 and Figure 9, which shows that the oscillations will increase further when ν is smaller and smaller.In contrast, our method still keeps stable.

Conclusions
In this article, a upwind finite volume element scheme has been derived for the two-dimensional time fractional coupled Burgers' equation.We discrete the equation using L1 formula in the time direction.We then discrete the resulting equations in space domain using upwind finite volume element method.The convergence analyses of semi-discrete scheme and fully discrete scheme are proved.Then, the numerical experiments are given to verify the effectiveness of the proposed scheme.And our method keeps stable when ν is smaller by contrast with finite volume element method without upwinding technique.Due to the influence of the linear scheme and the upwinding technique, the convergence order of the scheme is O(τ + h).In future works, we will use this method to solve more nonlinear fractional partial differential equations, such as nonlinear convection-diffusion equation and Navier-Stokes equation.And for the Caputo fractional derivative with α ∈ (0, 1), we will try to use other approximation methods (such as L1-2, L2-1 σ formulas) and high-precision upwinding technique to improve the convergence order.