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

Zhao, Jie; Li, Hong; Fang, Zhichao; Bai, Xue

doi:https://doi.org/10.1155/2020/6321209

Discrete Dynamics in Nature and Society

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Abstract Introduction Numerical Example Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 6321209 | https://doi.org/10.1155/2020/6321209

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

Jie Zhao,^1,2Hong Li ,¹Zhichao Fang,¹and Xue Bai¹

Academic Editor: Fabio Tramontana

Received22 Jul 2019

Accepted16 Jan 2020

Published19 Mar 2020

Abstract

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. The existence, uniqueness, and stability analyses for semidiscrete and fully discrete MFVE schemes are given in detail. The optimal a priori error estimates for the unknown and auxiliary variables in the norm are derived by using the stability results. Finally, numerical results are given to verify the feasibility and effectiveness.

1. Introduction

In this article, we consider the following one-dimensional Burgers’ equation:with initial and boundary conditionswhere , with , α is a positive constant, is the viscosity coefficient, and 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. The 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 [6–13], 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 one-dimensional Burgers’ equation and obtained the optimal error estimates for the velocity and flux (gradient) in the norm. Pany et al. [16] applied an -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 norm. Hu et al. [18] constructed a Crank–Nicolson time discretization MFE scheme to treat two-dimensional Burgers’ equation by using the 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 norm.

The 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 [22–25] with the finite volume element (FVE) methods [26–30]. The 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 [32–34], 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 norm. Furthermore, we give some numerical results to verify the feasibility and effectiveness of the MFVE scheme.

The 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. The 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 .

2. Semidiscrete MFVE Scheme

We introduce a flux function as an auxiliary variable, where . Then, we can rewrite equations (1) and (2) as the following first-order system:

The mixed variational formulation of (3) is to find such that

Now, we construct the primal mesh for the interval with the nodes , where N is some positive integer. Then, the primal mesh is denoted by , and the diameter of the primal mesh is defined by , where . We assume that the mesh satisfies the quasiuniform condition for some constant .

Next, the corresponding dual mesh is constructed by the nodes , where . Denote , , and , then the dual mesh is defined by .

We choose the mixed finite element space as the trial function space, where

Let and be the basis of the spaces and , respectively, where is the piecewise linear polynomial defined in [40] and is the characteristic function of the set .

The system (3) is integrated as follows:

Now, we define the transfer operator (see [40]) as follows:

Thus, we have . The range of is used as the test function space. Then, we can rewrite (6) as

Note thatBy a simple calculation, it is easy to have that , , . Then, we get the semidiscrete MFVE scheme to find such thatwhere satisfies

3. Theoretical Analysis for the Semidiscrete MFVE Scheme

3.1. Some Lemmas

For theoretical analysis, we first give some properties of the transfer operator and two projection operators.

Lemma 1 (see [40]). The transfer operator is bounded, that is,

Lemma 2 (see [40]). The transfer operator satisfies the following symmetry relation:

Lemma 3 (see [40]). The transfer operator satisfies the following positivity:

Lemma 4 (see [40]). Let I be an identity operator, then the transfer operator satisfies the following properties:

Lemma 5 (see [40]). There exists a constant such that

Now, the elliptic projection operator is introduced below, which satisfies

At the same time, the orthogonal projection operator is introduced to satisfy

Referring to References [24, 25], we can know that the projection operators and satisfy the following estimate properties.

Lemma 6. There exists a constant such that, for ,

3.2. Existence, Uniqueness, and Stability Analyses

Theorem 1. There exists a unique discrete solution for the semidiscrete MFVE scheme (10).

Proof. Obviously, there exists a unique solution for the scheme (11). Let and be the basis functions of the spaces and , respectively, then and can be expressed as follows:Choosing and in (10), we rewrite the semidiscrete scheme (10) in the matrix form to find such thatwhereIt is easy to know that matrices A and D are symmetrically positive definite. Then, the system (21) can be rewritten asAccording 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 be the discrete solution of the semidiscrete scheme (10), then there exists a constant such that

Proof. Choosing and in (10), we haveNoting that , we rewrite (25) asIntegrating (26) from 0 to t, we getApplying Lemma 1 and Lemma 3, we obtainCalculating the derivative of (10) with respect to t, we getSetting in (10) and in (29), we haveApplying the Sobolev embedding theorem to estimate the right-hand side of (30), we obtainMaking use of (31) and Lemma 3 in (30), we haveIntegrating (32) from 0 to t, we obtainApply the Gronwall lemma and (28) in (33), we obtainNext, choosing in (10), we getNoting that , we haveSubstituting (34) into the above inequality, we haveThus, applying the Sobolev embedding theorem, we obtainNow, we estimate and . We choose in (11) to obtainChoosing in (11), we haveMaking use of (39), we easily getThus, we obtain the proof of Theorem 2.

3.3. A Priori Error Estimates

We first rewrite the errors aswhere and are the elliptic projection operators and the orthogonal projection operator, respectively. By subtracting (10) from (4), we get thatwhere and satisfy

Theorem 3. Let and be the solutions of the systems (4) and (10), respectively. Assume that the initial solution satisfies (11), then there exists a constant such thatwhere represents the function of and .

Proof. Taking and in (43), we haveApplying Theorem 2, we have . We apply the Cauchy–Schwarz inequality and the Young inequality in (46) to obtainIntegrating (47) from 0 to t, and applying Lemma 3, we obtainApplying the Gronwall lemma in (48) yieldsNext, calculating the derivative of (43) with respect to t, we obtainChoosing in (50) and in (43), we haveApplying the stability results in Theorem 2, we havewhere represents the function of and . Applying Lemmas 3–5 in (51), and making use of (52), we obtainIntegrating (53) from 0 to t, we getApplying the Gronwall lemma in (54), we obtainWe estimate by choosing in (44) and noting that :Applying Lemma 6 in (56), we haveFinally, we apply Lemma 6 and the triangle inequality to obtain the desired conclusion.

4. Fully Discrete MFVE Scheme and Its Theoretical Analysis

4.1. Fully Discrete MFVE Scheme

Let be an equidistant partition of the time interval , where M is a positive integer. Denote to represent the step length and . And denote and for a function ψ.

Let and be the fully discrete solutions of u and p at , respectively. We can obtain the following nonlinear backward Euler MFVE scheme to find :where the initial value satisfies the following equations:

Remark 1. The fully discrete MFVE scheme (58) is implicit in time. In the actual calculation of , we need to make predictions first by using the linear backward Euler MFVE scheme defined as follows:Similar to the proof process of Theorem 4 in Reference [40], we can also obtain that there exists a unique discrete solution for the linear MFVE scheme (60).

4.2. Existence, Uniqueness, and Stability Analyses

For the existence analysis, we first give the Brouwer fixed-point theorem [41, 42].

Lemma 7. Let be a finite-dimensional inner product space with a norm . Let be a continuous operator. Assume that there exists such that for with . Then, there exists such that and .

Theorem 4. Suppose that has been given, then there exists a unique fully discrete solution for the nonlinear backward Euler MFVE scheme (58).

Proof. Choosing in (58), we haveThe operator is defined as follows:It is easy to know that the operator is continuous. Setting in (62), we haveSelecting to satisfy , we have . Making use of Lemma 7, we know that there exists such that . Thus, we choose to satisfy (61). Furthermore, selecting in (58), we getThus, there exists a solution which satisfies (64). Then, it is obviously known that 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 be another solution of the scheme (58) with the initial value , then we haveLet , then we haveThe following proof is based on the mathematical induction. First, ; next assume and then choose in (66) to obtainApplying the stability results in Theorem 5 (because the existence of the discrete solutions has been proved), we haveMultiplying (68) by , and making use of Lemma 3, we obtainSelecting to satisfy , we have and , which indicate that , and then is known from (58). Thus, we complete the proof of Theorem 4.

Theorem 5. Let and be the discrete solutions of the systems (58) and (59), respectively. Then, there exists a constant such thatFurthermore, the initial solution satisfies

Proof. Similarly to the proof process of Theorem 2, we can obtain the estimate of the initial solution . We choose and in (58) to obtainTaking note of and in (72), we obtainMultiplying (73) by , summing from 1 to n, and applying Lemma 1 and Lemma 3, we haveNext, we make use of (58) and (59) to obtainChoosing in (58) and in (75), we getNoting thatand applying Lemma 3 in (76), we haveSumming (78) from 1 to n and applying the inverse inequality and the Sobolev embedding theorem, we obtainSelecting and h in (79) to satisfy , making use of (74), and applying the discrete Gronwall lemma, we haveChoosing in (58), we getApplying the Cauchy–Schwarz inequality, we obtainMaking use of (80) in (82), and applying the Sobolev embedding theorem, we obtainThus, we complete the proof of Theorem 5.

4.3. A Priori Error Estimates

First, the errors are expressed as

By subtracting (58) from (4), we can obtain the error equations about and as follows:where and and satisfy

Theorem 6. Let be the discrete solution of the fully discrete scheme (58), and the exact solution of the mixed variational formulation (4) satisfies the following regularity properties:Then, there exists a constant such that

Proof. Choosing and in (85), we haveApplying Theorem 5, we get . Noting thatwe easily get thatMultiplying (91) by , and summing from 1 to m, we haveApplying Lemma 3 in (92), we obtainSelecting to satisfy , and applying the discrete Gronwall lemma, we obtainAccording to the integral remainder formula of Taylor formula, we getApplying the above inequalities and Lemma 6, we obtainNext, we estimate . Making use of (85) and (86), we haveThe right-hand side of (97) can be estimated by Theorem 5 as follows:where and . For the convenience of description, we use to denote the function of and .
Now, choosing in (85) and in (97), we haveApplying (98) and Lemmas 3–5, we obtainMultiplying (100) by , and summing from 1 to m, we getSelecting to satisfy , and applying the discrete Gronwall lemma, we haveWe estimate by choosing in (86):Applying Lemma 6, we haveTaking note of in (102), and applying (95), (96), and Lemma 6, we obtainFinally, we apply the triangular inequality and Lemma 6 to complete the proof of Theorem 6.

5. Numerical Example

Now, we give a numerical example to verify the feasibility and effectiveness of the MFVE scheme. We consider Burgers’ equation as follows:where . Following Reference [4], we can obtain the exact solution (Fourier series solution) as follows:where

The auxiliary variable is determined by the above data.

In practical calculation, we use the equidistant grids of spatial regions and truncate the series of the exact solutions with when calculating error results in the norm of u and p. We give the error results and convergence orders in Table 1 with mesh sizes . We can see that the convergence orders are approximate to 1, and these results are consistent with Theorem 6. The space-time graphs of the exact solutions of u and p at with are shown in Figures 1 and 2, respectively. The space-time graphs of the nonlinear backward Euler fully discrete solutions and at with are shown in Figures 3 and 4. The figures show that the numerical solutions of u and p have the same numerical behavior as their exact solutions. The numerical results and figures verify that the MFVE method for Burgers’ equation is feasible and efficient.

6. Conclusions

In this article, our main aim is to construct the MFVE scheme for solving Burgers’ equation by introducing the auxiliary variable and the transfer operator . 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 norm for the velocity u and flux p by using the stability results in Theorems 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

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the draft of the manuscript and read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11761053 and 11701299), the Natural Science Foundation of Inner Mongolia Autonomous Region (2016BS0105 and 2017MS0107), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-17-A07), and the Prairie Talent Project of Inner Mongolia Autonomous Region.

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Copyright © 2020 Jie Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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