# A New Linear Difference Scheme for Generalized Rosenau-Kawahara Equation

Authors: Tao Chen; Kaili Xiang; Peimin Chen; Xumei Luo
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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 inL∞-norm of the difference scheme by an induction argument and the discrete energy method, 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.


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## Body

## 1. Introduction

In the study of the dynamics of dense discrete systems, Rosenau in [1, 2] proposed the so-called Rosenau equation:(1)ut+ux+uxxxxt+uux=0,as the KdV equation cannot describe the phenomena of wave-wave and wave-wall interactions. In [3], the existence and the uniqueness of the solution of (1) have been studied. For further consideration of nonlinear waves, adding the viscous terms uxxx and -uxxxxx to (1), respectively, gives the following equations:(2)ut+uxxxxt+ux+αuxxx+γupx=0,(3)ut+uxxxxt+ux+αuxxx-βuxxxxx+γupx=0,where α, β, and γ are all real constants, while for the exponent, we assume that p≥2 is an integer. Equations (2) and (3) are called the generalized Rosenau-RdV equation [4–6] and the generalized Rosenau-Kawahara equation [7, 8]. When p=2, they can be considered as Rosenau-KdV equation and the Rosenau-Kawahara equation, respectively.As we all know, most of the time, we need to think of the numerical solution of nonlinear evolution equations. Many scholars in this field have carried out good work. In [9–22], the nonlinear Crank-Nicolson scheme (see [9, 10, 13, 16, 19, 21]) and the linear finite difference scheme (see [11–15, 18, 20, 22]) have been used for the generalized Rosenau-type equations, including the Rosenau-Burgers equation, the Rosenau-RLW equation, the Rosenau-KdV equation, and the Rosenau-Kawahara equation. Those classical finite difference schemes often use the formula (up)x=p/1+pup-1ux+upx to construct second-order convergent linear finite difference scheme [11–15, 18, 20, 22]. In [23], the authors used the formula u2x=2θuux+1-θu2x and proposed a linear finite difference scheme, but the proof of the prior estimate in L∞-norm is not perfect.In this paper, using techniqueUjnpx=p3/2Ujn-1/2Ujn-1p-1Ujn+1/2x^, we propose a semiexplicit linearized Crank-Nicolson finite difference method for the generalized Rosenau-Kawahara equation (3) with an initial condition (4)ux,0=u0x,x∈xL,xR,and boundary conditions(5)uxL,t=uxR,t=0,uxL,tx=uxR,tx=0,uxL,txx=uxR,txx=0,t∈0,T.By comparison with the classical second-order accuracy finite difference scheme on a test problem [15, 20–22], our new scheme improves the CPU time and gives a better maximal error en∞ of numerical solutions. But the prior estimate in L∞-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 [25–27]) 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 L∞-norm of the difference scheme by an induction argument and the discrete energy method, and then we obtain the prior estimate in L∞-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.


## 2. Finite Difference Schemes

In this section, we give a complete description of our numerical method for the initial value problem (3)–(5). We first give some notation which will be used in this paper. As usual, we let J, N be any positive integers. Let h=xR-xL/J, τ=T/N; the domain of solution is defined to be Ωhτ=Ωh×Ωτ, which is covered by uniform grid Ωh=xj=xL+jh;j=0,1,…,J, Ωτ=tn=nτ;n=0,1,…,N. Suppose U=Ujn∣j=0,1,…,J,n=0,1,…,N is a discrete function on Ωhτ and denote uxj,tn≡ujn,uxj,tn≈Ujn, Zh0=U=Uj∣U-2=U-1=U0=UJ=UJ+1=UJ+2=0,j=-2,-1,0,1,…,…,J,J+1,J+2, C as a general positive constant independent of step sizes h and τ which may have different values at different occurrences. Introduce the following notations: (6)Ujnx=Uj+1n-Ujnh,Ujnx¯=Ujn-Uj-1nh,Ujnx^=Uj+1n-Uj-1n2h,Ujnt=Ujn+1-Ujnτ,Ujn+1/2=Ujn+1+Ujn2,Un2=Un,Un,Un,Vn=h∑j=0J-1UjnVjn,Un∞=max0≤j≤J-1⁡Ujn.In this paper, we propose a new linear finite difference scheme for the generalized Rosenau-Kawahara equation (3)–(5) which is written as(7)Ujnt+Ujnxxx¯x¯t+Ujn+1/2x^+αUjn+1/2xx¯x^-βUjn+1/2xxx¯x¯x^+p3/2Ujn-1/2Ujn-1p-1Ujn+1/2x^=0,Uj0=u0xj,j=0,1,2,…,JUn∈Zh0,U0nx^=UJnx^,U0nxx¯=UJnxx¯,n=0,1,2,…,N.This scheme is a semiexplicit linearized Crank-Nicolson scheme; the truncation error of this scheme is of order Oh2+τ2. In this scheme, complicated nonlinear term (Ujn)px is extrapolated by p3/2Ujn-1/2Ujn-1p-1Ujn+1/2x^. Thus, we only need to solve a linear system of equations in computing Ujn+1. Hence, scheme (7) can be expected to be more efficient.


## 3. Convergence and Stability

In this section, we prove the convergence and stability of scheme (7). Let ejn=ujn-Ujn, where ujn and Ujn are the solutions of (3)–(5) and (7), respectively. We then obtain the following error equation:(8)rjn=ejnt+ejnxxx¯x¯t+ejn+1/2x^+αejn+1/2xx¯x^-βejn+1/2xxx¯x¯x^+Rjn,where(9)Rjn=p3/2ujn-1/2ujn-1p-1ujn+1/2x^-p3/2Ujn-1/2Ujn-1p-1Ujn+1/2x^,and rjn denotes the truncation error. By using Taylor expansion at xj,tn+1/2, we easily obtain that the truncation error of scheme satisfies(10)rjn=Oh2+τ2,τ,h⟶0.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 functionsU,V∈Zh0, one has (11)Ux,V=-U,Vx¯,Ux^,V=-U,Vx^,Uxx¯,V=-Ux,Vx,and then one has (12)U,Uxx=-Ux,U,Ux^=0,U,Uxx¯x^=0,U,Uxxx¯x¯x^=0.Furthermore, if U0xx=UJxx=0, then (13)U,Uxxx¯x¯=Uxx,2Un+1/2,Uxxx¯x¯t=1τUxx¯n+12-Uxx¯n2.The following lemmas including Lemmas2 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 2 (discrete Sobolev inequality [24]).
For any discrete functionU=Ujn∣j=0,1,…,J,n=0,1,…,N on the finite interval xL,xR, there is the inequality (14)Un∞≤C0UnUxn+Un,or(15)Un∞≤εUxn+KεUn.Here, ε, C0, and K(ε) are three constants independent of  U=Ujn∣j=0,1,…,J,n=0,1,…,N and τ, h. ε can be any small and Kε is a constant dependent on ε.Lemma 3 (discrete Gronwall’s inequality [24]).
Suppose that the discrete functionωn∣n=0,1,…,N;Nτ=T satisfies the inequality (16)ωn≤A+τ∑l=1nBlωl,where A and Bl(l=0,1,…,N) are nonnegative constants. Then, (17)max1≤n≤N⁡ωn≤Aexp⁡2τ∑l=1NBl,where τ is sufficiently small, such that τ·max1≤n≤N⁡Bl≤1/2.Lemma 4 (see [22]).
Suppose thatu0∈H02xL,xR. Then, the solution of problems (3)–(5) satisfies (18)uL2≤C,uxL2≤C,uxxL2≤C,uL∞≤C,uxL∞≤C.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 [25–27]). The following theorem shows the convergence of our scheme (7) with the convergence rate Oτ2+h2 in the L∞-norm by an induction argument.Theorem 5.
Supposeu0∈H02xL,xR, and ux,t∈Cx,t7,3; then the solution of the difference problem (7) converges to the solution of problem (3)–(5) with order Oτ2+h2 in the L∞-norm, if (19)32C0·max⁡Cn-1,Cnτ2+h2≤1,τ≤12p+13pp-1Cu+1p-1.Proof.
We use the mathematical induction to prove it. First, from (10) and Lemma 4, we have (20)rjn∞≤Cr,ujn∞≤Cu,n=0,1,2,…,N,where Cr and Cu are two constants independent of τ and h. It follows from the initial conditions that (21)e0=0,exx0=0,U0∞=0.We also can get U1 by the C-N scheme. Hence, the following estimate holds:(22)e1+exx1≤τ2+h2C1.Now, assume that (23)el+exxl≤τ2+h2Cl,l=0,1,2,…,n,0≤n≤N,where Cl is a constant independent of τ and h. Using Lemma 2 and Cauchy-Schwarz inequality, we get(24)el∞≤C0elexl+el≤12C02el+exl≤32τ2+h2C0Cl,l=0,1,2,…,n,0≤n≤N,Ul∞≤ul∞+el∞≤Cu+32τ2+h2C0Cl,l=0,1,2,…,n,0≤n≤N.Now, computing the inner product of error equation (8) with en+1/2 and using boundary condition (5) and Lemma 1, we obtain(25)12τen+12-en2+12τexxn+12-exxn2=rn,en+1/2-Rn,en+1/2=rn,en+1/2-ph∑j=1J-132Ujn-12Ujn-1p-1ejn+1/2x^ejn+1/2-ph∑j=1J-132ejn-12ejn-1∑k=0p-232ujn-12ujn-1p-2-k·32Ujn-12Ujn-1kujn+1/2x^ejn+1/2.Using Lemma 1, Lemma 4, Cauchy-Schwarz inequality, and (24) for (26)32C0·max⁡Cn-1,Cnτ2+h2≤1,we have(27)-ph∑j=1J-132Ujn-12Ujn-1p-1ejn+1/2x^ejn+1/2≤p2Cu+3τ2+h2C0·max⁡Cn-1,Cnp-1h∑j=1J-1ejn+1/2x^·ejn+1/2≤p2p-2Cu+1p-1en+1/22+exn+1/22≤p2p-4Cu+1p-13en+12+3en2+exxn+12+exxn20≤n≤N-1,(28)-ph∑j=1J-132ejn-12ejn-1∑k=0p-232ujn-12ujn-1p-2-k·32Ujn-12Ujn-1kujn+1/2x^ejn+1/2≤p∑k=0p-22Cup-2-k2Cu+3τ2+h2C0·max⁡Cn-1,Cnk·Cuh∑j=0J-132ejn-12ejn-1·ejn+1/2≤pp-12p-2CuCu+1p-2h∑j=0J-132ejn-12ejn-1·ejn+1/2≤pp-12p-4Cu+1p-13en2+4en+1/22+en-12≤pp-12p-4Cu+1p-12en+12+5en2+en-12,0≤n≤N-1.On the other hand,(29)rn,en+1/2=12rn,en+1+en≤12rn2+14en+12+en2.Substituting (27)-(29) into (25), we obtain(30)en+12-en2+exxn+12-exxn2≤τrn2+τ2en+12+en2+τp2p-3Cu+1p-13en+12+3en2+exxn+12+exxn2+τpp-12p-3Cu+1p-12en+12+5en2+en-12≤τrn2+τpp-12pCu+1p-1·en+12+en2+en-12+exxn+12+exxn2,0≤n≤N-1.Summing up (30) from 0 to N-1, we get(31)eN2+exxN2≤e02+exx02+τ∑n=0N-1rn2+τ∑n=0N3pp-12pCu+1p-1en2+exxn2.Taking small τ and h such that (32)τ∑k=0N-1rn2≤τNmax1≤n≤N⁡rn2≤Tτ2+h22Cr2,then using (22) and Lemma 3, for (33)τ≤13pp-12p+1Cu+1p-1,we have (34)eN2+exxN2≤τ2+h22Cr2Te2T3pp-12pCu+1p-1≤τ2+h22Cn+12,n=0,1,2,…,N-1,where Cn+1=CrTe3pp-1T2pCu+1p-1 is a constant independent of n. By an induction argument, (35)en≤Oτ2+h2,exxn≤Oτ2+h2,n=0,1,2,…,N.From Cauchy-Schwarz inequality, we can obtain (36)exn≤Oτ2+h2,n=0,1,2,…,N.By using Lemma 2, it is shown that (37)en∞≤Oτ2+h2,n=0,1,2,…,N.This completes the proof of Theorem 5.The following theorem guarantees that the numerical solution obtained from the difference scheme (7) is bounded. This can be proved from (24) of Theorem 5.Theorem 6.
Supposeu0∈H02xL,xR, and ux,t∈Cx,t7,3; if τ and h are small enough, then, for n=1,2,…,N, the inequality (38)Ujn∞≤C~holds, where C~ is a constant independent of τ and h.By a similar proof of Theorem5, we can obtain the following theorems. Theorem 7 shows that the solution of difference scheme (7) is stable in L∞-norm. Theorem 8 guarantees the existence and uniqueness of numerical solution.Theorem 7.
Under the conditions of Theorem5, the solution of scheme (7) is stable in L∞-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 (39)Ujnt+Ujnxxx¯x¯t+Ujn+1/2x^+αUjn+1/2xx¯x^-βUjn+1/2xxx¯x¯x^+p32Ujnp-1-12Ujn-1p-1Ujn+1/2x^=0,Uj0=u0xj,j=0,1,2,…,JUn∈Zh0,U0nx^=UJnx^,U0nxx¯=UJnxx¯,n=0,1,2,…,N.Its second-order convergence in theL∞-norm and stability can be proved by a similar proof to that of scheme (7) in this paper.


## 4. 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 Ujn+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) asNew. In [15], we denote the linear difference scheme asLinear 1. In [20], we denote the linear difference scheme asLinear 2 when p=2. In [22], we denote the linear difference scheme asLinear 3 when p=8. We will measure the accuracy of the proposed scheme using the maximum norm errors defined by e∞=un-Un∞. The second-order convergence of the numerical solutions is verified directly from enh,τ∞/enh/2,τ/2∞.


### 4.1. Example 1

Consider the Rosenau-KdV equation with the initial condition(40)u0x=-3548+35624313sech4⁡124-26+2313xand the boundary conditions (41)uxL,t=uxR,t=0,uxL,tx=uxR,tx=0,uxL,txx=uxR,txx=0,0≤t≤T.It is known that the solitary wave solution [4] is (42)u0x=-3548+35624313sech4⁡124-26+2313x-12+126313t.The results in terms of the maximal norm errors and CPU time at the timeT=40 using α=1, β=0, xL=-80, xR=120, and p=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 L∞-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 α=1, β=0, p=2, and τ=h=1/40. Clearly, our proposed new scheme gives smaller maximal error than the scheme in [15].Table 1
Comparison of the maximal errorse∞ and CPU time for the Rosenau-KdV equation at t=40 with α=1, β=0, and p=2.
Scheme h = τ = 1 / 10 h = τ = 1 / 20 h = τ = 1 / 40 e ∞ CPU(s) e ∞ CPU(s) e ∞ CPU(s) Linear 1[15] 9.39 e - 4 4.11 2.35 e - 4 14.85 5.91 e - 5 69.06 New 4.66 e - 4 4.02 1.17 e - 4 14.07 2.95 e - 5 66.66Table 2
The numerical verification of theoretical accuracyO(h2+τ2) for Rosenau-KdV equation at different time at t=40 with α=1,β=0, and p=2.
t e n h , τ ∞ / e n h / 2 , τ / 2 ∞ h = τ = 1 / 10 h = τ = 1 / 20 h = τ = 1 / 40 t = 10 - 3.9730964 4.0264475 t = 20 - 3.9871826 4.0049147 t = 30 - 3.9940428 3.9816829 t = 40 - 3.9982009 3.9577326Figure 1
Comparison between maximal errors of numerical solutions computed by scheme [15] and new scheme for Rosenau-KdV equation with α=1, β=0, h=τ=1/40, and p=2.


### 4.2. Example 2

According to [7, 8], when α=β=1, p=2, the solitary wave solution of the initial boundary problem (3)–(5) is (43)u0x=-3524+35312205sech4⁡112-13+205x-113+205t,and when α=β=1, p=8, the solitary wave solution [7] is (44)ux,t=-351224-85+75491/7sech4/7⁡736-85+7549x-754985t.The results in terms of the maximal norm errors and CPU time at the timeT=40 using xL=-80, xR=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 e∞ 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].Table 3
Comparison of the maximal errorse∞ and CPU time for Rosenau-Kawahara equation at t=40 with α=1, β=1, and p=2.
Scheme h = τ = 1 / 10 h = τ = 1 / 20 h = τ = 1 / 40 e ∞ CPU(s) e ∞ CPU(s) e ∞ CPU(s) Linear 2 [20] 2.51 e - 4 6.01 6.28 e - 5 22.60 1.56 e - 5 100.43 New 1.19 e - 4 5.87 3.00 e - 5 22.45 7.44 e - 6 97.91Table 4
Comparison of the maximal errorse∞ and CPU time for generalized Rosenau-Kawahara equation at t=40 with α=1, β=1, and p=8.
Scheme h = τ = 1 / 10 h = τ = 1 / 20 h = τ = 1 / 40 e ∞ CPU(s) e ∞ CPU(s) e ∞ CPU(s) Linear 3 [22] 1.01 e - 3 7.47 2.53 e - 4 28.84 6.62 e - 5 124.68 New 3.97 e - 4 6.89 9.77 e - 5 26.32 2.84 e - 5 114.59Table 5
The numerical verification of theoretical accuracyOh2+τ2 for generalized Rosenau-Kawahara equation.
t p = 2 p = 8 h = 1 / 10 h = 1 / 20 h = 1 / 40 h = 1 / 10 h = 1 / 20 h = 1 / 40 t = 10 - 3.9760462 4.0631134 - 4.0350242 3.6676336 t = 20 - 3.9878186 4.0811327 - 4.0699563 3.5392528 t = 30 - 3.9915570 4.0556430 - 4.0749519 3.7479758 t = 40 - 3.9933012 4.0316195 - 4.0701304 3.5400894Figure 2
Comparison between maximal errors of numerical solutions computed by scheme [20] and new scheme for Rosenau-Kawahara equation with α=1, β=1, h=τ=1/40, and p=2.Figure 3
Comparison between maximal errors of numerical solutions computed by scheme [22] and new scheme for generalized Rosenau-Kawahara with α=1, β=1, h=τ=1/40, and p=8.


## 4.1. Example 1

Consider the Rosenau-KdV equation with the initial condition(40)u0x=-3548+35624313sech4⁡124-26+2313xand the boundary conditions (41)uxL,t=uxR,t=0,uxL,tx=uxR,tx=0,uxL,txx=uxR,txx=0,0≤t≤T.It is known that the solitary wave solution [4] is (42)u0x=-3548+35624313sech4⁡124-26+2313x-12+126313t.The results in terms of the maximal norm errors and CPU time at the timeT=40 using α=1, β=0, xL=-80, xR=120, and p=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 L∞-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 α=1, β=0, p=2, and τ=h=1/40. Clearly, our proposed new scheme gives smaller maximal error than the scheme in [15].Table 1
Comparison of the maximal errorse∞ and CPU time for the Rosenau-KdV equation at t=40 with α=1, β=0, and p=2.
Scheme h = τ = 1 / 10 h = τ = 1 / 20 h = τ = 1 / 40 e ∞ CPU(s) e ∞ CPU(s) e ∞ CPU(s) Linear 1[15] 9.39 e - 4 4.11 2.35 e - 4 14.85 5.91 e - 5 69.06 New 4.66 e - 4 4.02 1.17 e - 4 14.07 2.95 e - 5 66.66Table 2
The numerical verification of theoretical accuracyO(h2+τ2) for Rosenau-KdV equation at different time at t=40 with α=1,β=0, and p=2.
t e n h , τ ∞ / e n h / 2 , τ / 2 ∞ h = τ = 1 / 10 h = τ = 1 / 20 h = τ = 1 / 40 t = 10 - 3.9730964 4.0264475 t = 20 - 3.9871826 4.0049147 t = 30 - 3.9940428 3.9816829 t = 40 - 3.9982009 3.9577326Figure 1
Comparison between maximal errors of numerical solutions computed by scheme [15] and new scheme for Rosenau-KdV equation with α=1, β=0, h=τ=1/40, and p=2.


## 4.2. Example 2

According to [7, 8], when α=β=1, p=2, the solitary wave solution of the initial boundary problem (3)–(5) is (43)u0x=-3524+35312205sech4⁡112-13+205x-113+205t,and when α=β=1, p=8, the solitary wave solution [7] is (44)ux,t=-351224-85+75491/7sech4/7⁡736-85+7549x-754985t.The results in terms of the maximal norm errors and CPU time at the timeT=40 using xL=-80, xR=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 e∞ 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].Table 3
Comparison of the maximal errorse∞ and CPU time for Rosenau-Kawahara equation at t=40 with α=1, β=1, and p=2.
Scheme h = τ = 1 / 10 h = τ = 1 / 20 h = τ = 1 / 40 e ∞ CPU(s) e ∞ CPU(s) e ∞ CPU(s) Linear 2 [20] 2.51 e - 4 6.01 6.28 e - 5 22.60 1.56 e - 5 100.43 New 1.19 e - 4 5.87 3.00 e - 5 22.45 7.44 e - 6 97.91Table 4
Comparison of the maximal errorse∞ and CPU time for generalized Rosenau-Kawahara equation at t=40 with α=1, β=1, and p=8.
Scheme h = τ = 1 / 10 h = τ = 1 / 20 h = τ = 1 / 40 e ∞ CPU(s) e ∞ CPU(s) e ∞ CPU(s) Linear 3 [22] 1.01 e - 3 7.47 2.53 e - 4 28.84 6.62 e - 5 124.68 New 3.97 e - 4 6.89 9.77 e - 5 26.32 2.84 e - 5 114.59Table 5
The numerical verification of theoretical accuracyOh2+τ2 for generalized Rosenau-Kawahara equation.
t p = 2 p = 8 h = 1 / 10 h = 1 / 20 h = 1 / 40 h = 1 / 10 h = 1 / 20 h = 1 / 40 t = 10 - 3.9760462 4.0631134 - 4.0350242 3.6676336 t = 20 - 3.9878186 4.0811327 - 4.0699563 3.5392528 t = 30 - 3.9915570 4.0556430 - 4.0749519 3.7479758 t = 40 - 3.9933012 4.0316195 - 4.0701304 3.5400894Figure 2
Comparison between maximal errors of numerical solutions computed by scheme [20] and new scheme for Rosenau-Kawahara equation with α=1, β=1, h=τ=1/40, and p=2.Figure 3
Comparison between maximal errors of numerical solutions computed by scheme [22] and new scheme for generalized Rosenau-Kawahara with α=1, β=1, h=τ=1/40, and p=8.


## 5. 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 inL∞-norm of the difference scheme and then obtain the prior estimate in L∞-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 equations, Ginzburg-Landau equation, and a generalization of the BBM equation [28].



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