Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods

In this article, mixed finite volume element (MFVE) methods are proposed for solving the numerical solution of Burgers’ equation. By introducing a transfer operator, semidiscrete and fully discrete MFVE schemes are constructed. (e existence, uniqueness, and stability analyses for semidiscrete and fully discrete MFVE schemes are given in detail. (e optimal a priori error estimates for the unknown and auxiliary variables in the L2(Ω) norm are derived by using the stability results. Finally, numerical results are given to verify the feasibility and effectiveness.


Introduction
In this article, we consider the following one-dimensional Burgers' equation: with initial and boundary conditions where Ω � (a, b), Γ � (0, T] with 0 < T < ∞, α is a positive constant, ] > 0 is the viscosity coefficient, and u 0 (x) is the given initial function.
Burgers' equation is a famous nonlinear evolution equation which was first derived by Bateman [1] in 1915. Burgers [2] utilized this equation to model turbulence behavior. Since its appearance, this equation has been widely concerned by researchers because of its various practical applications, such as gas dynamics, shock theory, traffic flows, viscous flow, and turbulence. e exact solutions can be expressed as a Fourier series expansion by introducing a Hopf-Cole transformation [3,4]. Benton and Platzman [5] gave the exact solutions of Burgers' equation in one-dimensional spatial regions for some different initial functions. In the past several decades, many numerical techniques had been constructed and tested to solve Burgers' equation [6][7][8][9][10][11][12][13], such as finite element methods, finite difference methods, least-squares finite element methods, and spectral methods.
In recent years, the mixed finite element (MFE) methods and finite volume (FV) methods have been used to solve Burgers' equation by many researchers. Luo and Liu [14] proposed an MFE method to solve one-dimensional Burgers' equation by introducing a flux function as an auxiliary variable and gave the existence, uniqueness, and error analyses for the discrete solutions. Chen and Jiang [15] constructed a characteristic MFE scheme to solve onedimensional Burgers' equation and obtained the optimal error estimates for the velocity and flux (gradient) in the L 2 norm. Pany et al. [16] applied an H 1 -Galerkin MFE method to approximate the velocity and flux of one-dimensional Burgers' equation and gave a priori error estimates and numerical experiments. Shi et al. [17] provided a low-order least-squares nonconforming characteristic MFE scheme to solve two-dimensional Burgers' equation and obtained the optimal order error estimates in the L 2 norm. Hu et al. [18] constructed a Crank-Nicolson time discretization MFE scheme to treat two-dimensional Burgers' equation by using the P 2 0 − P 1 pair and gave the optimal error analysis and numerical experiments. Nascimento et al. [19] applied a Fourier pseudospectral method and an FV method to solve one-dimensional Burgers' equation and gave the numerical comparison. Guo et al. [20] proposed a fifth-order FV weighted compact scheme to solve one-dimensional Burgers' equation and gave numerical experiments. Sheng and Zhang [21] proposed an FV method to solve two-dimensional Burgers' equation and obtained the optimal error estimate in the H 1 norm. e aim of this article is to develop mixed finite volume element (MFVE) methods to solve one-dimensional Burger's equation by combining the MFE methods [22][23][24][25] with the finite volume element (FVE) methods [26][27][28][29][30]. e MFVE methods, also called mixed covolume methods, were first proposed by Russell [31] to solve the elliptic equation. Now, the methods have been applied to solve second-order elliptic equations [32][33][34], integrodifferential equations [35], parabolic equations [36,37], time-fractional partial differential equations [38], and so on. In this article, we introduce a flux function as an auxiliary variable, rewrite (1) as the first-order system, and construct the semidiscrete and nonlinear backward Euler fully discrete MFVE schemes by using a transfer operator. We give the theoretical analysis for semidiscrete and fully discrete schemes in detail, including existence, uniqueness, and stability. In particular, in the analysis of the fully discrete scheme, we apply the Brouwer fixed-point theorem to prove the existence and use the Sobolev embedding theorem and the inverse inequality to prove the stability. Making use of the stability results of discrete solutions, we obtain the optimal a priori error estimates for the velocity and flux in the L 2 norm. Furthermore, we give some numerical results to verify the feasibility and effectiveness of the MFVE scheme. e rest of this article is organized as follows: In Section 2, we use a flux function as an auxiliary variable and give the mixed variational formulation and the semidiscrete MFVE scheme. e existence, uniqueness, stability, and convergence analyses for semidiscrete and fully discrete schemes are given in Section 3 and 4, respectively. In Section 5, a numerical example is given to verify the theoretical results. In this article, the standard definitions and notations of the Sobolev spaces as in [39] are used. Furthermore, we use the symbol C to represent a generic constant which is independent of the space and time mesh parameters h and Δt.

Semidiscrete MFVE Scheme
We introduce a flux function p(x, t) � (α/2)f(u(x, t)) − ]u x (x, t) as an auxiliary variable, where f(u) � u 2 . en, we can rewrite equations (1) and (2) as the following first-order system: e mixed variational formulation of (3) is to find Now, we construct the primal mesh for the interval Ω � Next, the corresponding dual mesh is constructed by the nodes a � , then the dual mesh is defined by I * h � A * i : i � 0, 1, · · · , N . We choose the mixed finite element space H 0h × L h as the trial function space, where Let ϕ i : i � 1, 2, · · · , N − 1 and χ A i : i � 0, 1, · · · , N − 1 be the basis of the spaces H 0h and L h , respectively, where ϕ i is the piecewise linear polynomial defined in [40] and χ A i is the characteristic function of the set A i . e system (3) is integrated as follows: Now, we define the transfer operator c h : H 0h ⟶ L 2 (Ω) (see [40]) as follows: Discrete Dynamics in Nature and Society . e range Y h of c h is used as the test function space. en, we can rewrite (6) as Note that By a simple calculation, it is easy

Some Lemmas.
For theoretical analysis, we first give some properties of the transfer operator c h and two projection operators.
Lemma 1 (see [40]). e transfer operator c h is bounded, that is, Lemma 2 (see [40]). e transfer operator c h satisfies the following symmetry relation: Lemma 3 (see [40]). e transfer operator c h satisfies the following positivity: Lemma 4 (see [40]). Let I be an identity operator, then the transfer operator c h satisfies the following properties: Lemma 5 (see [40]).
Now, the elliptic projection operator At the same time, the L 2 orthogonal projection operator Referring to References [24,25], we can know that the projection operators Π h and R h satisfy the following estimate properties.

Lemma 6.
ere exists a constant C > 0 such that, for i � 0, 1, Discrete Dynamics in Nature and Society

Existence, Uniqueness, and Stability Analyses
ere exists a unique discrete solution for the semidiscrete MFVE scheme (10).
Proof. Obviously, there exists a unique solution be the basis functions of the spaces H 0h and L h , respectively, then u h ∈ H 0h and p h ∈ L h can be expressed as follows: Choosing (10), we rewrite the semidiscrete scheme (10) It is easy to know that matrices A and D are symmetrically positive definite. en, the system (21) can be rewritten as According to the theory of differential equation, we can see that the system (23) has a unique solution, which shows that there exists a unique discrete solution for the semidiscrete MFVE scheme (10). □ Theorem 2. Let u h , p h be the discrete solution of the semidiscrete scheme (10), then there exists a constant C > 0 such that Proof.
Choosing v h � u h and q h � u hx in (10), we have Noting that (f(u h ), u hx ) � 0, we rewrite (25) as 1 2 Integrating (26) from 0 to t, we get Applying Lemma 1 and Lemma 3, we obtain Calculating the derivative of (10) with respect to t, we get Setting v h � u ht in (10) Applying the Sobolev embedding theorem to estimate the right-hand side of (30), we obtain α ] Making use of (31) and Lemma 3 in (30), we have Integrating (32) from 0 to t, we obtain Apply the Gronwall lemma and (28) in (33), we obtain Next, choosing q h � u hx in (10), we get 4 Discrete Dynamics in Nature and Society Noting that (f(u h ), u hx ) � 0, we have Substituting (34) into the above inequality, we have us, applying the Sobolev embedding theorem, we obtain Now, we estimate ‖u h (0)‖ 1 and ‖p h (0)‖. We choose z h � u hx (0) in (11) to obtain Choosing q h � p h (0) in (11), we have Making use of (39), we easily get us, we obtain the proof of eorem 2.

A Priori Error Estimates.
We first rewrite the errors as where Π h and R h are the elliptic projection operators and the L 2 orthogonal projection operator, respectively. By subtracting (10) from (4), we get that where η(0) and ε(0) satisfy (10), respectively. Assume that the initial solution u h (0), p h (0) satisfies (11), then there exists a constant C > 0 such that

Fully Discrete MFVE Scheme and Its
Theoretical Analysis where M is a positive integer. Denote Δt � T/M to represent the step length and t n � nΔt, n � 0, 1, . . . , M. And denote ψ n � ψ(t n ) and z t ψ n � (ψ n − ψ n− 1 )/Δt for a function ψ. Let u n h and p n h be the fully discrete solutions of u and p at t � t n , respectively. We can obtain the following nonlinear backward Euler MFVE scheme to find u n h , p n h ∈ H 0h × L h (n � 1, 2, . . . , M): where the initial value u 0 h , p 0 h satisfies the following equations: Discrete Dynamics in Nature and Society Remark 1.
e fully discrete MFVE scheme (58) is implicit in time. In the actual calculation of u n h , p n h (1 ≤ n ≤ M), we need to make predictions first by using the linear backward Euler MFVE scheme defined as follows: Similar to the proof process of eorem 4 in Reference [40], we can also obtain that there exists a unique discrete solution for the linear MFVE scheme (60).

Existence, Uniqueness, and Stability Analyses.
For the existence analysis, we first give the Brouwer fixed-point theorem [41,42].
It is easy to know that the operator g is continuous.
Selecting V ∈ H 0h to satisfy ‖V‖ � 1 + (4 � 3 √ /3)‖u n− 1 h ‖, we have (g(V), V) H 1 > 0. Making use of Lemma 7, we know that there exists V * ∈ H 0h such that g(V * ) � 0. us, we choose u n h � V * to satisfy (61). Furthermore, selecting u n h � V * in (58), we get us, there exists a solution p n h ∈ L h which satisfies (64). en, it is obviously known that u n h , p n h satisfies the scheme (58), which proves the existence of the fully discrete solutions.
Next, we give the uniqueness for the fully discrete scheme (58). Let V n , Q n { } ∈ H 0h × L h (n � 1, 2, . . . , M) be another solution of the scheme (58) with the initial value V 0 � u 0 h , then we have e following proof is based on the mathematical induction. First, E 0 � u 0 h − V 0 � 0; next assume E n− 1 � 0 and then choose v h � E n in (66) to obtain Applying the stability results in eorem 5 (because the existence of the discrete solutions has been proved), we have Multiplying (68) by Δt, and making use of Lemma 3, we obtain Proof. Similarly to the proof process of eorem 2, we can obtain the estimate of the initial solution u 0 h , p 0 h . We choose v h � u n h and q h � u n hx in (58) to obtain Taking note of (f(u n h ), u n hx ) � 0 and Multiplying (73) by 2Δt, summing from 1 to n, and applying Lemma 1 and Lemma 3, we have Next, we make use of (58) and (59) to obtain Choosing v h � z t u n h in (58) and q h � p n h in (75), we get Noting that Applying the above inequalities and Lemma 6, we obtain Finally, we apply the triangular inequality and Lemma 6 to complete the proof of eorem 6.
In practical calculation, we use the equidistant grids of spatial regions and truncate the series of the exact solutions with 500 n�1 when calculating error results in the L ∞ (L 2 (Ω)) norm of u and p. We give the error results and convergence orders in Table 1 with mesh sizes h � Δt � 1/20, 1/40, 1/80, 1/160. We can see that the convergence orders are approximate to 1, and these results are consistent with eorem 6. e space-time graphs of the exact solutions of u and p at t � 2 with h � 1/40 are shown in Figures 1 and 2

Conclusions
In this article, our main aim is to construct the MFVE scheme for solving Burgers' equation by introducing the auxiliary variable p(x, t) and the transfer operator c h . We give the detailed theoretical results on existence, uniqueness, and stability for the semidiscrete and fully discrete schemes and obtain the optimal a priori error estimates in the L 2 (Ω) norm for the velocity u and flux p by using the stability results in eorems 2 and 5. Moreover, we give a numerical example to verify that the proposed scheme for Burgers' equation is feasible and efficient. In the future, we will apply the MFVE methods to solve some other nonlinear partial differential equations.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.