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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 351619, 11 pages
http://dx.doi.org/10.1155/2013/351619
Research Article

The Upwind Finite Volume Element Method for Two-Dimensional Burgers Equation

School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China

Received 15 October 2012; Accepted 8 January 2013

Academic Editor: Xiaodi Li

Copyright © 2013 Qing Yang. 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.

Abstract

A finite volume element method for approximating the solution to two-dimensional Burgers equation is presented. Upwind technique is applied to handle the nonlinear convection term. We present the semi-discrete scheme and fully discrete scheme, respectively. We show that the schemes are convergent to order one in space in -norm. Numerical experiment is presented finally to validate the theoretical analysis.

1. Introduction

We consider the following two-dimensional Burgers equation [13]: for the unknown functions and in a bounded spatial domain , over a time interval . The coefficient is a positive number.

Burgers equation is the simplest nonlinear convection-diffusion model [1]. It is often used in modeling such physical phenomena as turbulence, shocks, and so forth. The study of Burgers equation has been a very active area because of its importance.

It is well known that strictly parabolic discretization schemes applied to Burgers equation do not work well when it is advection dominated. Effective discretization schemes recognize to some extent the hyperbolic nature of the equation.

The finite volume element method (FVEM) [412] is an important discretization technique for partial differential equations, especially those that arise from physical conservation laws. FVEM has ability to be faithful to the physics in general and conservation in particular, to produce simple stencils, and to treat effectively Neumann boundary conditions and nonuniform grids, and so forth.

Liang [11, 12] combined the upwind technique and the FVEM to handle the linear convection-dominated problems. In this paper, we will consider upwind finite volume element method for the approximation of (1). Upwind approximation is applied to handle the nonlinear convection term. The semi-discrete and fully discrete schemes are defined, respectively. We prove that they are both convergent to order one in space. Numerical experiments are presented finally to validate the theoretical analysis.

In this paper, we use the following Sobolev spaces and the norms associated with these spaces: In particular, ,  . Let and let be any of the spaces just defined. If represents functions on , we set If , we drop it from the notation. We also drop ; thus, we write for .

If is a vector function, we say that if and .

An outline of the paper follows. In the next section we define the upwind finite volume element schemes for (1). Some Lemmas are presented in Section 3. We derive the -norm error estimates for the semi-discrete scheme and the fully discrete scheme in Sections 4 and 5, respectively. Finally in Section 6, we give some numerical experiments.

Throughout the paper we will denote by and generic constants independent of the mesh parameters, which may take different values in different occurrences.

2. The Approximation Schemes

In order to rewrite (1) as the vector form we define some vector notations. The gradient of a vector function is a matrix, and the divergence of a matrix function is a vector Consequently, we have for a vector function Let , , and let ; then the system (1) can be written as the following vector form: where

Let be a triangulation of the domain , and as usual, we assume the triangles to be shape regular. Denote by the set of the vertices of all the triangles , and let . For a given triangulation , we construct a dual mesh whose elements are called control volumes. Each triangle can be divided into three subdomains by connecting an inner point of the triangle to the midpoints of the three edges. Around each , we associate a control volume , which consists of the union of subregions having as a vertex. For a vertex , we can define its control volume in a similar way. Then we define the dual partition to be the union of all the control volumes. Usually we can choose the inner points as the barycenters or the circum centers, and in the later case we assume that all the inner angles of each triangle are not larger than . We will use the barycenters duel mesh in this paper, while, with some trivial changes, our analysis can be also applied to the case when the circum centers are used.

We now characterize the finite-dimensional spaces which will be employed in approximating (6). For the sake of simplicity, we will assume that . We define the following finite dimensional spaces: where denotes the set of polynomials on with a degree of not more than .

Multiplying (6a) by test function and integrating by parts yield where here is the unit outward normal vector of .

Now we approximate by using the upwind technique.

Let is adjoint with . Assuming that , let and is the length of . Denote by the unit outward normal vector of when is regarded as the boundary of . Define Let The upwind discretization of the nonlinear term is defined by the form Using the heaviside function we can write as

Introduce the interpolation operators and , respectively. For , define and . Assuming that , we can easily get the following interpolation estimates:

The semi-discrete upwind finite volume scheme of (6) is as follows: find such that where is the interpolation projection of , that is, .

Partition into , with . Our analysis is valid for variable time steps, but we drop the superscript from for convenience. For functions on , we write for . By approximating at the time with the backward difference , we define the fully discrete upwind finite volume scheme for (6) as follows: find , such that

3. Some Lemmas

Now we present several lemmas. Let .

Lemma 1.    is a self-adjoint operator, that is,
Let . Then, for some positive constants and that are independent of ,

Proof. It is easy to know that From [4] we know that for , , , , where and are some positive constants that are independent of . Thus we obtain (19) and (20) immediately.

Lemma 2. For the bilinear form , one has the following conclusions:(i)For , one has (ii)There exists a positive constant such that (iii)There exists a positive constant such that

Proof. For , define the bilinear form Then, we get By combining the above results and the corresponding conclusions for in [4], we can obtain (23)–(25).

Lemma 3. For , and , one has

Proof. First we have Noting that , , we can easily deduce by the definition of . Now we only need to bound We denote the last two terms on the right-hand side of (31) by and , respectively. We now turn to analyze the two terms. Noting that , we rewrite as here is the set of vertex of . From the Taylor’s Formula and the linear property of , we obtain that
Applying the trace inequality, we get Similarly, we can deduce that We conclude that
The similar argument yields the estimate
Substituting the estimates (36) and (37) into (31), we obtain This yields the desired result immediately.

4. Error Bounds for Semi-Discrete Scheme

Theorem 4. Assume that and are solutions to (6) and (17), respectively. also assumes that is regular enough. Then there exists a positive constant such that where depends on principally , , and .

Proof. We derive the following error equation from (6) and (17): Let . We rewrite the previously mentioned equation as We choose in (41) to get Using Lemmas 1, 2 and Young’s inequality, we have Now we bound the last two terms on the right-hand side of (43). We need the following induction hypothesis:
We know from [13] that This implies that Also we have the following inverse inequality:
Using (46), we get Next, we write By Choosing , and in Lemma 3, using (47) and the Young’s inequality, we can obtain By an argument like (36) and then by (46) and (47), we have Substituting (50) and (51) into (49), we get Make (43), (48), and (52) together to obtain Integrating the previously mentioned equation from 0 to and noting (44), we obtain that for sufficiently small and . By using Lemma 2(ii) and the Gronwall’s inequality, we have that It follows from the interpolation estimates that
Now we prove the induction hypothesis (44). Noting that , we know that (44) holds obviously for . It follows from (56) that Then (44) holds for any .
By (16), we have From the triangle inequality, we obtain where depends on , , and . We now complete the proof of the theorem.

5. Error Bound for the Fully Discrete Scheme

Theorem 5. Assume that satisfies the necessary regularities and the discretization parameters obey the relation . Then the error of the approximation (18) of (6) satisfies where depends on , , , and .

Proof. Subtract (18) from (9) to obtain that Choose to obtain that For the left-hand side of (62), from Lemmas 1 and 2, we have We denote terms on the right-hand side of (62) by . Then, (62) can be rewritten as
Now we estimates the terms one by one. From the Taylor’s formula, we have It follows that For the next two terms, we have
We make the following induction hypothesis:
For , using the similar argument as (48) and noting (68), we deduce that
Now, we write and can be handled as and in Theorem 4. Thus, we have
Substituting the previously mentioned estimates into (64), we get
Multiplying (72) by and summing over , we have By choosing and small enough and noting Lemma 2(ii), we have
Applying the Gronwall inequality and the interpolation theory, we deduce that
Now we prove the induction hypothesis (68). Noting that , we know that . From (75) and the assumption , we get that Thus we know that (68) holds for any . Using triangular inequality and the interpolation theory completes the proof.

6. Numerical Example

In this section, we will show the affectivity of our method by numerical experiments. The exact solutions to problem (1) can be obtained by employing Cole-Hopf transformation. For , we consider the following solutions: where . We present numerical results for the -norm estimates of and . In Tables 1 and 2, we present the numerical results for and , respectively. In all runs, we use the uniform mesh step and choose the time . As seen in these tables, in all cases the errors decrease by a factor of about two as decreases by the factor of two. This indicates that all -norm error estimates are of first-order convergence, which is consistent with our theoretical analysis.

tab1
Table 1: Numerical results for .
tab2
Table 2: Numerical results for .

When and , the figures of the exact solutions and the numerical solutions at for are given in Figures 1, 2, 3, and 4. In order to show that our method keeps stable when is smaller, we also give the comparison figures of exact solution and numerical solution for in Figures 5 and 6. The comparison figure of numerical solution by using finite volume element method (FVEM) without upwinding is given in Figure 7, which show that the approximation produces unacceptable nonphysical oscillations.

351619.fig.001
Figure 1: The exact solution when at .
351619.fig.002
Figure 2: The numerical solution when at , for .
351619.fig.003
Figure 3: The exact solution when at .
351619.fig.004
Figure 4: The numerical solution of when at , for .
351619.fig.005
Figure 5: The exact solution at at .
351619.fig.006
Figure 6: The numerical solution by FVEM with upwinding when at .
351619.fig.007
Figure 7: The numerical solution by FVEM without upwinding when at .

Acknowledgments

This paper is supported by The Natural Science Foundation of China (Grant nos. 10971254 and 11171193) and the Natural Science Foundation of Shandong Province (Grant nos. ZR2009AZ003 and ZR2011AM016).

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