Abstract and Applied Analysis

Volume 2013 (2013), Article ID 768976, 12 pages

http://dx.doi.org/10.1155/2013/768976

Research Article

Explicit Multistep Mixed Finite Element Method for RLW Equation

Yang Liu,¹ Hong Li,¹ Yanwei Du,¹ and Jinfeng Wang²

¹School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

²School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China

Received 15 February 2013; Revised 22 April 2013; Accepted 30 April 2013

Academic Editor: Marco Donatelli

Copyright © 2013 Yang Liu 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.

Abstract.

An explicit multistep mixed finite element method is proposed and discussed for regularized long wave (RLW) equation. The spatial direction is approximated by the mixed Galerkin method using mixed linear space finite elements, and the time direction is discretized by the explicit multistep method. The optimal error estimates in and norms for the scalar unknown and its flux based on time explicit multistep method are derived. Some numerical results are given to verify our theoretical analysis and illustrate the efficiency of our method.

1. Introduction

In this paper, we consider the following initial boundary problem of RLW equation:

where is a bounded open interval, with . The initial value is given function, and the coefficients , , are all positive constants.

In recent years, large nonlinear phenomena are found in many research fields, for example, physics, biology, fluid dynamics, and so forth. These phenomena can be described by the mathematical model of some nonlinear evolution equations. In particular, some attention has also been paid to nonlinear RLW equations [1, 2] which play a very important role in the study of nonlinear dispersive waves. Solitary waves are wave packet or pulses, which propagate in nonlinear dispersive media. Due to dynamical balance between the nonlinear and dispersive effects, these waves retain an unchanged waveform. A soliton is a very special type of solitary wave, which also keeps its waveform after collision with other solitons. The regularized long wave (RLW) equation is an alternative description of nonlinear dispersive waves to the more usual Korteweg-de Vries (KdV) equation. Mathematical theories and numerical methods for (1) were considered in [1–24]. The existence and uniqueness of the solution of RLW equation are discussed in [5]. Their analytical solutions were found under restricted initial and boundary conditions, and therefore they got interest from a numerical point of view. Several numerical methods for the solution of the RLW equation have been introduced in the literature. These include a variety of difference methods [5–8, 17], finite element methods based on Galerkin and collocation principles [9–13], mixed finite element methods [14–16], meshfree method [18], adomian decomposition method [19], and so on.

In [25], Chatzipantelidis studied the explicit multistep methods for some nonlinear partial differential equations and discussed some mathematical theories. Akrivis et al. [26] studied the multistep method for some nonlinear evolution equations. Mei and Chen [20] presented the explicit multistep method based on Galerkin method for regularized long wave (RLW) equation. In this paper, our purpose is to propose and study an explicit multistep mixed method, which combines a mixed Galerkin method in the spatial direction and the explicit multistep method in the time direction, for RLW equation. We derive optimal error estimates in and norms for the scalar unknown and its flux for the fully discrete explicit multistep mixed scheme and compare our method’s accuracy with some other numerical schemes. Compared to the numerical methods in [20, 25, 26], we not only obtain the approximation solution for , but also get the approximation solution for .

The layout of the paper is as follows. In Section 2, an explicit multistep mixed scheme and numerical process are given. The optimal error estimates in and norms for the scalar unknown and its flux for the fully discrete explicit multistep mixed scheme are proved in Section 3. In Section 4, some numerical results are shown to confirm our theoretical analysis. Finally, some concluding remarks are given in Section 5. Throughout this paper, will denote a generic positive constant which does not depend on the spatial mesh parameter or time discretization parameter .

2. The Mixed Numerical Scheme

With the auxiliary variable , we reformulate (1) as the following first-order coupled system:

We consider the following mixed weak formulation of (2). Find satisfying:

Noting the Dirichlet boundary conditions and , we can get easily and then get the scheme (4).

Let and be finite dimensional subspaces of and , respectively, defined by

where is a partition of into subintervals , , , , and denotes the polynomials of degree less than or equal to in .

The semidiscrete mixed finite element method for (3) and (4) consists in determining such that

In the following discussion, we will give an explicit multistep mixed scheme. We take linear finite element spaces and , and then and can be expressed as the following formulation:

Substitute (8) into (6) and (7), and take and in (6) and (7), respectively, to obtain

where .

We subdivide the space variable domain into uniform subintervals with grid points , such that , . Using the local coordinate transformation , , we transform a subinterval into a standard interval . Furthemore, we have

We take linear basis functions defined as follows:

and then the variables and over the element are written as

Then, we get the following equations:

Then, the system (13) has the following matrix form:

with the following element matrices:

Assembling contributions from all elements, we obtain the following coupled system of nonlinear matrix equations:

To formulate a fully discrete scheme, we consider a uniform partition of with time step length , , and time levels , . We now discuss a fully discrete scheme based on a linear explicit multistep method. We now define and as approximations to and , respectively, and formulate the following fully discrete linear explicit multistep mixed scheme:

with given the initial approximations and . In the explicit multistep mixed system (17), the parameter variable and is described by the coefficients of the term , for the following polynomials and , respectively:

In this paper, we consider the explicit 2-step mixed method for the RLW equation. For , we obtained easily

Substituting (19) into (17), we obtain the following 2-step mixed scheme:

Remark 1. There have been many numerical schemes for the RLW equation, but we have not seen the related research on explicit multistep mixed element method for RLW equation in the literature. From the viewpoint of numerical theory, we propose a mixed element scheme (6) and (7), which is different from some other mixed finite element methods in [14–16], for the RLW equation and derive some a priori error estimates based on the explicit multistep mixed element method. From the perspective of numerical calculation, our method is efficient for RLW equation.

3. Two-Step Mixed Scheme and Optimal Error Estimates

3.1. Two-Step Mixed Scheme and Some Lemmas

In this section, we will discuss some a priori error estimates based on explicit 2-step mixed finite element method for the RLW equation. For the fully discrete procedure, let   be a given partition of the time interval with step length , for some positive integer . For a smooth function on , define .

The system (3) and (4) has the following formulation at :

Based on system (17), we get an equivalent formulation for system (21) as

where

We now find a pair in satisfying

For the theoretical analysis of a priori error estimates, we define the following projections.

Lemma 2 (see [15, 27, 28]). One defines the elliptic projection by

With , the following estimates are well known for :

Lemma 3 (see [15, 27, 28]). Furthermore, one also defines a Ritz projection of as the solution of

where , and is taken appropriately so that

where is a positive constant. Moreover, it is easy to verify that is bounded.

With , the following estimates hold:

For fully discrete error estimates, we now write the errors as

Combine (27), (29), (22), (23), (25), and (26) at to get the following error equations:

where

Lemma 4. For , , , and , the following estimates hold:

Proof. Using the Taylor expansion, we have

Combining (37) and (38) and noting that , we obtain

From (39), we have

Using a similar estimate as the one for , we have

where , .

From (41), we have

Using the Taylor expansion and noting that , we have

Using (43), we obtain

By (44), we have

Using the similar method to the estimate for and (31), we obtain

3.2. Optimal Error Estimates

In this subsection, we derive the fully discrete optimal error estimates and obtain the following theorem.

Theorem 5. Assuming that , and , are given, then for , , one has

Proof. Taking in (33) and using Cauchy-Schwarz’s inequality and Poincaré’s inequality, we get

Set in (34) to obtain

Use (48) as well as the Cauchy-Schwarz and Young’s inequalities to obtain

Note that

We now estimate , , , and as

Substituting (53) into (52), we obtain

Substituting (36), (51), and (54) into (50), using (48), and summing from , the resulting inequality becomes

Choose in such a way that for , . Then, as an application of Gronwall’s lemma, we obtain

Combine (28), (31), and (56) with the triangle inequality to complete the and error estimates for . Furthermore, use (48) and the triangle inequality to complete the optimal error estimates for and .

Remark 6. Compared to a variety of difference methods in [17], our method is studied based on mixed element scheme (6) and (7).

Remark 7. Although some convergence proofs of multistep methods for RLW/BBM are provided in [20, 25, 29], our convergence results of multistep methods are proved based on a mixed finite element scheme. Based on the current discussion, we have to provide the detailed proofs for multistep mixed finite element methods in this paper.

4. Numerical Results

In order to test the viability of the proposed method, we consider a test problem. We write conservation laws as [3, 4]

where , , and are usually called mass, momentum, and energy, respectively, which are observed to check the conservation of the numerical scheme.

We consider RLW equation (1) and let, in (1), . Then, the solitary wave solution of (1) is

where

We consider the motion of a single solitary wave and take as initial condition, with and ,

The corresponding exact solution with initial condition (60) is

In this procedure, we take space-time domain as and .

In Table 1, we take spatial mesh parameter and time discretization parameter and list the three invariants , , and and the optimal error estimate in and norms for at different times , , , , , and . At the same time, we show some numerical results at time obtained by other numerical methods in Table 1. From Table 1, we find that our method is more accurate than the numerical methods in [17, 18, 28] but is less than the numerical methods in [23, 24]. From the shown results in Table 1, we can see that , and keep almost constants, so the conservation for our method is very well.

Table 1: Solitary wave Amp. 0.3 and the errors in and norms for , , , and at , , , and .

Method
Time

for
for

Our method
0
3.9797
0.8104
2.5787
0
0

4
3.9797
0.8104
2.5786

8
3.9797
0.8104
2.5786

12
3.9797
0.8104
2.5787

16
3.9795
0.8104
2.5787

20
3.9790
0.8103
2.5785

[20]
20
3.9800
0.8104
2.5792

[21]
20
3.9820
0.8087
2.5730

[22]
20
3.9905
0.8235
2.6740

—

[23]
20
3.9616
0.8042
2.5583

[24]
20
3.9821
0.8112
2.5813

In Tables 2 and 3, we take spatial mesh parameter and obtain the optimal error results in and norms for at different times , , , , , and with different time discretization parameters , and . From Tables 2 and 3, we see easily that the convergence rate for time is close to order 2. Similarly, the results for are shown in Tables 4 and 5.

Table 2: Convergence order and error in norm for of time with and .

Time

Order
Order

4

1.9151
1.9745

8

1.9906
2.0129

12

2.0188
2.0241

16

2.0360
2.0295

20

2.0486
2.0321

Table 3: Convergence order and error in norm for of time with and .

Time

Order
Order

4

1.9919
1.9403

8

2.0023
1.9653

12

2.0000
1.9801

16

2.0059
1.9724

20

1.9788
1.8907

Table 4: Convergence order and error in norm for of time with and .

Time

Order
Order

4

1.9229
1.9863

8

2.0024
2.0161

12

2.0336
2.0255

16

2.0532
2.0303

20

2.0677
2.0338

Table 5: Convergence order and error in norm for of time with and .

Time

Order
Order

4

2.0884
2.1107

8

2.0799
2.1201

12

2.0797
2.1397

16

2.0795
2.1381

20

2.0661
2.1414

In Tables 6 and 7, the optimal error results in and norms for at different times , , , , , and with different spatial mesh parameters , , and and time discretization parameter are shown. It is easy to see that the convergence rate for space is close to order 2. The similar results for are listed in Tables 8 and 9.

Table 6: Convergence order and error in norm for of space with and .

Time

Order
Order

4

1.9940
1.8982

8

2.0195
1.9746

12

2.0102
2.0008

16

2.0589
1.9978

20

2.0358
1.8265

Table 7: Convergence order and error in norm for of space with and .

Time

Order
Order

4

1.8967
1.9147

8

1.9850
1.9863

12

2.0178
2.0079

16

2.0372
2.0158

20

2.0497
2.0139

Table 8: Convergence order and error in norm for of space with and .

Time

Order
Order

4

2.0969
2.1218

8

2.0932
2.1350

12

2.0900
2.1436

16

2.0879
2.1505

20

2.0857
2.1501

Table 9: Convergence order and error in norm for of space with and .

Time

Order
Order

4

1.9305
1.9928

8

2.0156
2.0219

12

2.0500
2.0313

16

2.0722
2.0356

20

2.0882
2.0404

Figure 1 shows the surface for the exact solution in space-time domain (), and the corresponding surface for the numerical solution with and is described in Figure 2. From Figures 1 and 2, we see easily that the exact solution is approximated very well by the numerical solution . In Figures 3 and 4, we show the surface for the exact solution and the numerical solution , respectively, with and and obtain a good approximation solution for the exact solution .

Figure 1: Surface for exact solution .

Figure 2: Surface for numerical solution .

Figure 3: Surface for exact solution .

Figure 4: Surface for numerical solution .

The comparison between the exact solution and the numerical solution is described at different times , and with and in Figure 5. The similar comparison for the exact solution and the numerical solution is shown in Figure 6. Figures 5 and 6 show that the solitary wave for and moves to the right with unchanged form and velocity, respectively. Furthermore, the exact solutions and are approximated well by the numerical solutions and , respectively.

Figure 5: Comparison between and at times , , , , and with and .

Figure 6: Comparison between and at times , , , and with and .

In Figure 7, we show the comparison between and at different spaces , and with and to verify the efficiency for the proposed scheme in this paper. Figure 8 describes a similar result for .

Figure 7: Comparison between and at spaces , , , and with and .

Figure 8: Comparison between and at spaces , , , and with and .

From the previous analysis in Tables 1–9 and Figures 1–8, we can see that the numerical results confirm the theoretical results of Theorem 5 and our method is efficient for RLW equation.

5. Concluding Remarks

In this paper, we propose and analyze an explicit multistep mixed finite element method, which combines spatial mixed finite element method and time explicit multistep method, for RLW equation. We discuss the numerical process for our method, prove the theoretical results for the fully discrete explicit multistep mixed scheme, obtain the optimal convergence order, and compare our method’s accuracy with some other numerical schemes. Compared with the numerical method in [20, 25, 26], our method can obtain the optimal error estimates in and norms for the scalar unknown and its flux simultaneously. From our numerical results, we can see that our method is efficient for RLW equation.

Acknowledgments

This work was supported by the National Natural Science Fund of China (11061021), the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011 and NJZY13199), the Natural Science Fund of Inner Mongolia Province (2012MS0108 and 2012MS0106), and the Program of Higher-Level Talents of Inner Mongolia University (125119).

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