Advances in Mathematical Physics

Advances in Mathematical Physics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 217393 | 11 pages | https://doi.org/10.1155/2014/217393

Two Conservative Difference Schemes for Rosenau-Kawahara Equation

Academic Editor: Ricardo Weder
Received14 Sep 2013
Revised22 Dec 2013
Accepted07 Jan 2014
Published18 Mar 2014

Abstract

Two conservative finite difference schemes for the numerical solution of the initialboundary value problem of Rosenau-Kawahara equation are proposed. The difference schemes simulate two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference schemes are of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

1. Introduction

In the study of compact discrete systems, the wave-wave and wave-wall interactions cannot be described by the well-known KdV equation. To overcome this shortcoming of KdV equation, Rosenau proposed the following so-called Rosenau equation in [1, 2]: which is usually used to describe the dense discrete system and simulate the long-chain transmission model through an L-C flow in radio and computer fields. Park proved the existence and uniqueness of solution to (1) in [3]. However, it is difficult to find its analytical solution. Therefore, the numerical study of (1) is very significant and attract many scholars (see, e.g., [410]).

As the further consideration of nonlinear wave, Zuo obtained Rosenau-Kawahara equation by adding viscous term and to Rosenau equation (1) and studied the solitary solution and periodic solution of this equation in [10]. In [11], Labidi and Biswas got the integral of Rosenau-Kawahara equation by using He’s principle. Then the solitary solution and two invariance of a generalized Rosenau-Kawahara equation are investigated in [12]. To our best knowledge, there is no study on the numerical method of Rosenau-Kawahara equation. Therefore, we will study the difference approximate solution of Rosenau-Kawahara equation with the initial data and boundary conditions as follows: where is a known constant and is a smooth function. When , the solitary wave solution of (2) is (see [10]) The initial boundary value problem (2)–(4) is in accordance with the Cauchy problem of (2) when , . Hence, the boundary condition (4) is reasonable. It is easy to verify that (2)-(4) satisfy the following conservative quantities (see [12]): where , are both constants only depending on initial data.

It is well known that a reasonable difference scheme has not only high-accuracy but also can maintain some physical properties of original problem. Lots of numerical experiments show that conservative difference scheme can simulate the conservative law of initial problem well since it could avoid the nonlinear blow-up [1324]. Moreover, it is suitable to compute for long-time. Li and Vu-Quoc pointed in [14] that in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation. Therefore, constructing the conservative difference scheme is a significant job.

The rest of this paper is organized as follows. We respectively propose a two-level nonlinear Crank-Nicolson difference scheme and three-level linear difference scheme for initial boundary value problems (2)–(4) in Sections 2 and 3. We analyze its two discrete conservative laws. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference schemes are of second-order convergence and unconditionally stable. In Section 4, we verify our theoretical analysis by numerical experiments.

2. Nonlinear Crank-Nicolson Conservative Difference Scheme

In this section, we propose a two-level nonlinear Crank-Nicolson difference scheme and give the theoretical analysis.

In the rest of this paper, denotes a general positive constant which may denote different value in different occurrence.

2.1. Nonlinear Difference Scheme and Its Conservative Law

Let and be the uniform step size in the spatial and temporal direction, respectively. The interval is divided into equal parts, where is a fixed positive integer. Denote , , . Let be the difference approximation of at , that is, . Denote . We define the difference operators, inner product, and norms that will be used in this paper as follows: Consider the following finite difference scheme for problems (2)–(4): The discrete boundary condition (11) is reasonable from the boundary condition (4).

Lemma 1. For any two discrete functions , we have from summation by parts (see [22]). Thus, And if , then

The following theorem shows how the difference schemes (9)–(11) simulate the conservative law numerically.

Theorem 2. Suppose that ; then the difference schemes (9)–(11) are conservative for discrete energy; that is,

Proof. Multiplying in the two sides of (9) and summing up for from to , from boundary (11) and Lemma 1, we obtain From the definition of , (15) is deduced from (17).
Taking the inner of (9) with (i.e., ), from boundary (11) and Lemma 1, we get where . On the other hand, Substituting (19)–(22) into (18), we get From the definition of , we get (16) by deducing (23).

2.2. Solvability of the Difference Scheme

In order to prove the solvability of difference scheme, we present the following Brouwer fixed point theorem [25].

Lemma 3 (Brouwer fixed point theorem). Let be a finite dimensional inner product space; suppose that is continuous and there exists an such that for all with . Then there exists such that and .

Theorem 4. There exists that satisfies difference schemes (9)–(11).

Proof. We use the mathematical induction to prove our theorem.
Suppose that there exist that satisfy difference scheme (9) for . Next we prove that there exists that satisfies difference scheme.
Let be an operator on defined by Taking the inner product of (24) with , and noticing that we have
Therefore, for any , if , then . From Lemma 3, there exists such that . Choose . It is easy to verify that satisfies difference scheme (9).

2.3. Convergence, Stability, and Uniqueness of Solution

Suppose that is the solution to (2)–(4). Denote . Then the truncation error of difference schemes (9)–(11) is By Taylor expansion, we know that as .

Lemma 5. Suppose that . Then the solution of problems (2)–(4) satisfies

Proof. It follows from (7) that Using Hölder inequality and Cauchy-Schwarz inequality, we can get Thus, which also yields , from Sobolev inequality.

Lemma 6 (discrete Sobolev inequality [22]). There exist constants and such that

Lemma 7 (discrete Gronwall inequality [22]). Assume that , are nonnegative gridding functions, and is increasing. If , and for any then for any

Theorem 8. Suppose that . Then the solution of difference schemes (9)–(11) satisfies , , thus , .

Proof. From (16), we have From (13) and by Cauchy-Schwarz inequality Thus, Therefore, from Lemma 6, we obtain , .

Remark 9. Theorem 8 implies that the solution of difference schemes (9)–(11) is unconditionally stable in the sense of norm .

Theorem 10. Suppose that . Then the solution of difference schemes (9)–(11) converges to the solution of (2)–(4) in the sense of norm , and the convergent rate is .

Proof. Subtracting (9)–(11) from (27)–(29) and denoting , we have Taking the inner product of (40) with , we get Similar to (19)–(21), we have So, (43) can be rewritten as As from Lemma 5 and Theorem 8, we obtain Using Cauchy-Schwarz inequality, we have On the other hand, Substituting (48) and (49) into (45), we get Similar to the deduction process of (38), one can get Therefore, (50) is changed into Denoting , (52) is equivalent to that is, Choosing sufficiently small such that , we get Summing up (55) from to , we get Noticing that and . we have From Lemma 7, we get that is, Noticing equality (51), one can obtain From Lemma 6, we have

Similar to Theorem 10, we have the following theorems.

Theorem 11. The solution to difference schemes (9)–(11) is unique.

3. A Linear Conservative Difference Scheme

In this section, we propose a three-level linear conservative difference scheme for (2)–(4) and give the theoretical analysis.

3.1. Linear Difference Scheme and Its Conservative Law

Consider the following finite difference scheme for problems (2)–(4):

The discrete boundary condition (65) is reasonable from the boundary condition (4).

The following theorem shows how the difference schemes (63)–(65) simulate the conservative law as follows.

Theorem 12. Suppose that ; then the difference schemes (63)–(65) are conservative for discrete energy; that is,

Proof. Multiplying in both sides of (63) and summing up for , from boundary (65) and Lemma 1, we obtain Notice that
From the definition of , (66) is deduced from (68) and (69).
Taking the inner product of (63) with , from boundary (65) and Lemma 1, we obtain where . On the other hand, Substituting (71)–(74) into (70), we obtain From the definition of , we get (67) by deducing (75).

3.2. Solvability of the Difference Scheme

Theorem 13. There exists unique solution for difference schemes (63)–(65).

Proof. Use the mathematical induction to prove it. It is obvious that is uniquely determined by the initial condition (64). We also can get by (9)–(11). Now suppose is solved uniquely. Consider the equation of (63) for Taking the inner product of (76) with , from boundary condition (65) and Lemma 1, we get Similar to (71)–(73), we have Notice that Substituting (78)–(81) into (77), we get that is, (76) only admits zero solution. Therefore, there exists a unique that satisfies (63).

3.3. Convergence and Stability of the Difference Scheme

Suppose that is the solution to (2)–(4). Denote . Then the truncation error of difference schemes (63)–(65) is as follows: By Taylor expansion, we know that as .

Theorem 14. Suppose that . Then the solution of difference schemes (63)–(65) satisfies , , ; thus , .

Proof. From (67), we know that By (38), one can get Then, from Lemma 6, we get , .

Remark 15. Theorem 14 implies that the solution of difference schemes (63)–(65) is unconditionally stable in the sense of norm .

Theorem 16. Suppose that . Then the solution of difference schemes (63)–(65) converges to the solution of (2)–(4) in the sense of norm , and the convergent order is .

Proof. Subtracting (63)–(65) from (83)–(85) and letting , we have Taking the inner product of the two sides of (88) with , and using boundary condition (90) and Lemma 1, we obtain Similar to (71)–(73), we have Then (91) is changed into By Lemma 5 and Theorem 14, it is shown that Using Cauchy-Schwarz inequality, we have On the other hand, Substituting (95)-(96) into (93), we could obtain Using the deduce process which is similar to (38), we have Then (97) is changed into Let , (99) can be rewritten as follows: that is, By choosing that is small enough such that , then Summing up (102) from to , we obtain Notice that