A New Linear Difference Scheme for Generalized Rosenau-Kawahara Equation

We introduce in this paper a new technique, a semiexplicit linearized Crank-Nicolson finite difference method, for solving the generalized Rosenau-Kawahara equation.We first prove the second-order convergence in L∞-norm of the difference scheme by an induction argument and the discrete energymethod, and then we obtain the prior estimate in L∞-norm of the numerical solutions. Moreover, the existence, uniqueness, and satiability of the numerical solution are also shown. Finally, numerical examples show that the new scheme is more efficient in terms of not only accuracy but also CPU time in implementation.

By comparison with the classical second-order accuracy finite difference scheme on a test problem [15,[20][21][22], our new scheme improves the CPU time and gives a better maximal error ‖  ‖ ∞ of numerical solutions.But the prior estimate in  ∞ -norm of the numerical solutions is very hard to obtain directly; the proofs of convergence and stability are difficult for our new schemes (7).So, the discrete energy analysis method [24] and an induction argument (see [25][26][27]) are used to prove the second-order convergence and stability.Furthermore, our new method has a wide range of applications for the generalized Rosenau-type equations including the Rosenau-Burgers equation, the Rosenau-KdV equation, and the Rosenau-RLW equation.
The content of this paper is organized as follows.In the following, we propose a semiexplicit linearized Crank-Nicolson finite difference scheme for initial boundary value problems (3)- (5).In Section 3, we prove the second-order convergence in  ∞ -norm of the difference scheme by an induction argument and the discrete energy method, and then we obtain the prior estimate in  ∞ -norm of the numerical solutions.Moreover, based on the prior estimate, the existence and uniqueness of the numerical solution are also shown.Section 4 is devoted to the numerical tests of the new scheme and shows that our scheme has reliable accuracy and spends less CPU time than the classical schemes in implementation.Finally, we finish our paper by concluding remarks in the last section.

Convergence and Stability
In this section, we prove the convergence and stability of scheme (7).Let    =    −    , where    and    are the solutions of (3)-( 5) and (7), respectively.We then obtain the following error equation: where and    denotes the truncation error.By using Taylor expansion at (  ,  +1/2 ), we easily obtain that the truncation error of scheme satisfies The following lemma is a property of scheme (7); we can obtain that directly from the boundary conditions and notations.This is a well known result, which is essential for existence, uniqueness, convergence, and stability of our numerical solution.
Lemma 1 (see [18,24]).For any two discrete functions ,  ∈  0 ℎ , one has and then one has The following lemmas including Lemmas 2 and 3 are well known and useful for the proofs of the convergence and stability.Lemma 4 can be deduced directly from the Cauchy-Schwarz inequality and Sobolev inequality.
Lemma 4 (see [22]).Suppose that  0 ∈  2 0 [  ,   ].Then, the solution of problems ( 3)-( 5) satisfies It should be pointed out that the induction argument is very useful for proving the convergence of a difference scheme whose prior estimate is difficult to obtain directly (see [25][26][27]).The following theorem shows the convergence of our scheme (7) with the convergence rate ( 2 + ℎ 2 ) in the  ∞ -norm by an induction argument.
and (, ) ∈  7,3 , ; then the solution of the difference problem (7) converges to the solution of problem ( 3)-( 5) with order ( Proof.We use the mathematical induction to prove it.First, from (10) where   and   are two constants independent of  and ℎ.It follows from the initial conditions that       0      = 0, We also can get  1 by the C-N scheme.Hence, the following estimate holds: Now, assume that where   is a constant independent of  and ℎ.Using Lemma Now, computing the inner product of error equation ( 8) with  +1/2 and using boundary condition (5) and Lemma 1, we obtain Using Lemma 1, Lemma 4, Cauchy-Schwarz inequality, and (24) for we have − ℎ where  (36) By using Lemma 2, it is shown that         ∞ ≤  ( 2 + ℎ 2 ) , ( = 0, 1, 2, . . ., ) . (37) This completes the proof of Theorem 5.
By a similar proof of Theorem 5, we can obtain the following theorems.Theorem 7 shows that the solution of difference scheme (7) is stable in  ∞ -norm.Theorem 8 guarantees the existence and uniqueness of numerical solution.
Theorem 7.Under the conditions of Theorem 5, the solution of scheme (7) is stable in  ∞ -norm for the initial values.Theorem 8.There exists a unique solution for difference scheme (7).
Another similar semiexplicit scheme for the generalized Rosenau-Kawahara equation ( 3)-( 5) is written as Its second-order convergence in the  ∞ -norm and stability can be proved by a similar proof to that of scheme (7) in this paper.

Numerical Examples
In this section, we compute three numerical examples to demonstrate and validate the effectiveness of our difference scheme.As a test problem for the scheme proposed here, we chose three test problems for which exact solution or numerical solutions have been reported previously.For the Rosenau-KdV, Rosenau-Kawahara, and generalized Rosenau-Kawahara equations, the parameters used by other researchers [15,20,22] to obtain their results were taken as a guiding principle for our computations.As scheme (7) is a semiexplicit scheme, which is a linear system about  +1  , we use the Thomas algorithm to solve the system.All the numerical experiments were executed on a 3.20 GHz computer, with 8 G RAM, running Matlab 2013a.
For convenience, we denote the new linear difference scheme (7) as New.In [15], we denote the linear difference scheme as Linear 1.In [20], we denote the linear difference scheme as Linear 2 when  = 2.In [22], we denote the linear difference scheme as Linear 3 when  = 8.We will measure the accuracy of the proposed scheme using the maximum norm errors defined by ‖‖ ∞ = ‖  −  ‖ ∞ .The second-order convergence of the numerical solutions is verified directly from ‖  (ℎ, )‖ ∞ /‖  (ℎ/2, /2)‖ ∞ .
The results in terms of the maximal norm errors and CPU time at the time  = 40 using  = 1,  = 0,   = −80,   = 120, and  = 2 are reported in Table 1.It can be seen that the computational efficiency of the present new method is slightly better than that of the method in [15], in terms of grid point number.As shown in Table 2, the second-order convergence in  ∞ -norm of the new schemes verifies the correction of the theoretical analysis.In Figure 1, plots of maximal errors from the four schemes are presented when  = 1,  = 0,  = 2, and  = ℎ = 1/40.Clearly, our proposed new scheme gives smaller maximal error than the scheme in [15].

Example 2.
According to [7,8], when  =  = 1,  = 2, the solitary wave solution of the initial boundary problem The results in terms of the maximal norm errors and CPU time at the time  = 40 using   = −80,   = 120 are reported in Tables 3 and 4. It is clear from Tables 3 and  4 that results by our new method show improvement over the previous one reported by [20,22].As shown in Table 5, the second-order convergence of the numerical solutions is verified.In Figures 2 and 3, the graphs of the maximal errors ‖‖ ∞ are presented; the graphs show that our method gives  a better approximate solution than the scheme proposed in [20,22].Moreover, our proposed scheme gives less CPU time than [20,22].

Conclusion
In brief, we first proposed a semiexplicit linearized Crank-Nicolson finite difference scheme for generalized Rosenau-Kawahara equation.We prove the second-order convergence in  ∞ -norm of the difference scheme and then obtain the prior estimate in  ∞ -norm of the numerical solutions.The stability, existence, and uniqueness of the numerical solution are also shown.Finally, some examples were given to show the efficiency of the new scheme.For future research, our new method has a wide range of applications for some nonlinear wave equations including the generalized Rosenau-type By an induction argument,          ≤  ( 2 + ℎ 2 ) ,            ≤  ( 2 + ℎ 2 ) , ( = 0, 1, 2, . . ., ) .