A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation
aa r X i v : . [ m a t h . NA ] S e p A linearly implicit conservative di ff erence scheme for the generalizedRosenau-Kawahara-RLW equation Dongdong He a, ∗ , Kejia Pan b,c a School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China. b School of Mathematics and Statistics, Central South University, Changsha 410083, China. c Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Abstract
This paper concerns the numerical study for the generalized Rosenau-Kawahara-RLW equation obtained by couplingthe generalized Rosenau-RLW equation and the generalized Rosenau-Kawahara equation. We first derive the energyconservation law of the equation, and then develop a three-level linearly implicit di ff erence scheme for solving theequation. We prove that the proposed scheme is energy-conserved, unconditionally stable and second-order accurateboth in time and space variables. Finally, numerical experiments are carried out to confirm the energy conservation,the convergence rates of the scheme and e ff ectiveness for long-time simulation. Keywords:
Rosenau-Kawahara-RLW equation, finite di ff erence conservative scheme, convergence, stability. AMS subject classifications:
1. Introduction
The nonlinear wave is one of the most important scientific research areas. During the past several decades, manyscientists developed di ff erent mathematical models to explain the wave behavior, such as the KdV equation, the RLWequation, the Rosenau equation, and many others. In the following, we give a short review of these important wavemodels.The well known KdV equation u t + u xxx + uu x = , (1)was first introduced by Boussinesq [1] in 1877 and rediscovered by Diederik Korteweg and Gustav de Vries [2] in1895. Since then, there are a lot of studies on this equation and its variational form. Here we just mention some ofthe recent work. Kudryashov [3] reviewed the travelling wave solutions for the KdV and the KdV-Burgers equationsproposed by [4], Biswas [5] studied the solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coe ffi cients, while Wang et al. [6] investigated the solitons, shock waves for the potential KdV equation. Inaddition to the theoretical studies, readers can refer to [7, 8] for the numerical simulations of the KdV equation andthe generalized KdV equation.The regularized long-wave (RLW) equation (also known as Benjamin-Bona-Mahony equation) u t + u x + uu x − u xxt = , (2)was first proposed as a model for small-amplitude long wave of water in a channel by Peregrine [9, 10]. The regular-ized long-wave (RLW) equation and its di ff erent variational forms were well studied both theoretically and numeri-cally in the literature. Readers can refer to [11–17] for numerical studies and [18–20] for theoretical studies.When study the compact discrete systems, the well-known KdV equation can not describe the wave-wave andwave-wall interactions. To overcome the shortcoming of KdV equation, Rosenau proposed the following so-calledRosenau equation [21, 22]: u t + u xxxxt + u x + uu x = . (3) ∗ Corresponding author. E-mail address:[email protected](Dongdong He)
Preprint submitted to Applied Mathematics and Computation September 16, 2015 he existence and uniqueness of the solution for the Rosenau equation were theoretically proved by [23]. Besides thetheoretical analysis, numerical studies of equation (3) also exist in the literature, see [24–28] and references therein.For further consideration of nonlinear waves, the term − u xxt is included in the Rosenau equation. The resultingequation is usually called the Rosenau-RLW equation [29–31]: u t − u xxt + u xxxxt + u x + uu x = . (4)The above equation was further extended into the generalized Rosenau-RLW equation [32–34]: u t − u xxt + u xxxxt + u x + u m u x = , (5)where m ≥ ff erence scheme for the generalized Rosenau-RLW equation (5), however, the method isa nonlinear scheme, iterations are needed at each time step. Later on, Pan et al. [33] used a three-level linearly implicitconservative scheme for the same generalized Rosenau-RLW equation (5), where the method is a linear scheme andis also second-order convergent both in time and space variables.On the other hand, to consider another behavior of nonlinear waves, the viscous term u xxx needs to be included inthe Rosenau equation (3). The resulting equation is usually called the Rosenau-KdV equation [35–39]: u t + u xxxxt + u xxx + u x + uu x = . (6)For numerical investigations, Hu et al. [39] proposed a second-order linear conservative finite di ff erence method forthe Rosenau-KdV equation. However, numerical methods for the initial-boundary value problem of the Rosenau-KdVequation have not been studied widely.By coupling the above Rosenau-RLW equation and Rosenau-KdV equation, one can obtain the following Rosenau-KdV-RLW equation [40–44], u t − u xxt + u xxxxt + u xxx + u x + uu x = . (7)For numerical investigation, Wongsaijai et al. [40] proposed a three level implicit conservative finite di ff erence methodfor the above Rosenau-KdV-RLW equation. For theoretical studies, the solitary wave, shock waves, conservation lawsas well as the asymptotic behavior for the Rosenau-KdV-RLW equation with power law nonlinearity were studiedby [41–44], where the power law nonlinearity means the the last term in the left-hand side of equation (7) is replacedby a general nonlinear term ( u p ) x and p is any positive integer.In addition, the following Kawahara equation u t + u x + uu x + u xxx − u xxxxx = , (8)arose in the theory of shallow water waves with surface tension [45]. Equation (8) is called the modified Kawaharaequation if the third nonlinear term in the left-hand side is replaced by u u x . There is a wide range of literature on thenumerical investigations and theoretical studies for the usual Kawahara equation and the modified Kawahara equation.For theoretical aspects, some periodic and solitary wave solutions for both the Kawahara equation and the modifiedKawahara equation are provided in [46–48]. In addition to the theoretical studies, readers can refer to [49–51] for thenumerical studies of the Kawahara equation and the modified Kawahara equation.As one more step consideration of the nonlinear wave, Zuo [38] obtained the Rosenau-Kawahara equation byadding another viscous term − u xxxxx to the Rosenau-KdV equation (6), and studied the solitary solution and periodicsolution of the Rosenau-Kawahara equation. The Rosenau-Kawahara equation is given as follows [38]: u t + u x + uu x + u xxx + u xxxxt − u xxxxx = . (9)For numerical study, Hu et al. [52] proposed a two level nonlinear Crank-Nicolson scheme and another three-levelimplicit linear conservative finite di ff erence scheme for the Rosenau-Kawahara equation, both methods are proved to2e second-order convergent. And Biswas [53] investigated the solitary solution and the two invariance of followingequation, u t + au x + bu m u x + cu xxx + λ u xxxxt − ν u xxxxx = , (10)where a , b , c , ν, α, λ are real constants, m is a positive integer, which indicates the power law nonlinearity. Equation(10) is referred as the generalized Rosenau-Kawahara equation [52]. As far as we know, besides the analytical solitarysolution obtained by [53], there is no numerical study for this generalized Rosenau-Kawahara equation.Besides all the literature mentioned in the above, there are also a lot of related references which used both an-alytical and numerical methods to study di ff erent nonlinear models. For example, Triki et al. [54] discussed thesolitary solution for the Gear-Grimshaw model, Bhrawy et al. [55] provided solitons, cnoidal waves and snoisalwaves for the Whitham-Broer-Kaup system, while Shokri et al. [58] numerically solved the two-dimensional complexGinzburg-Landau equation by using the meshless radial basis functions method, and Dehghan et al. [59] applied asemi-analytical method for solving the Rosenau-Hyman equation. Readers can refer to [54–57] for more discussionson analytically finding solitons and other wave solutions for di ff erent nonlinear models, and refer to [58–60] for morediscussions on numerical studies.In this paper, we will study the following generalized Rosenau-Kawahara-RLW equation: u t + au x + bu m u x + cu xxx − α u xxt + λ u xxxxt − ν u xxxxx = , x l ≤ x ≤ x r , ≤ t ≤ T . (11)which is obtained by coupling the generalized Rosenau-RLW equation (5) and the generalized Rosenau-Kawaharaequation (10). Analytical solitary solutions of the above equation (11) were recently obtained by [61] through thesech ansatz method.In this work, we shall consider the numerical simulation of above equation (11) with initial condition u (0 , x ) = u ( x ) , x l ≤ x ≤ x r , (12)and boundary conditions u ( x l , t ) = u ( x r , t ) = , u x ( x l , t ) = u x ( x r , t ) = , u xx ( x l , t ) = u xx ( x r , t ) = , ≤ t ≤ T , (13)where m is a positive integer, a , b , c , ν are all real constants, while α, λ are set to be positive constants, x l is a largenegative number and x r is a large positive number. In this paper, we only discuss the solitary solution of equation(11). By solitary wave assumptions, the solitary solution and its derivatives have the following asymptotic values: u → x → ±∞ , and ∂ n u ∂ x n → x → ±∞ , for n ≥
1. Thus, the boundary conditions (13) are meaningful for thesolitary solution of equation (11). And we assume that the wave peak is initially located at x =
0, and x l , x r , which arelarge numbers, are used to assure that the solitary wave peak always locates inside the domain [ x l , x r ] during the timeinterval [0 , T ]. Similar set-ups are used in [39, 40, 52].Note that equation (11) reduces to the generalized Rosenau-RLW equation when c = , ν =
0. Specially, it reducesto the usual Rosenau-RLW equation (4) when a = m = α = λ = , b = , c = , ν =
0. And equation (11) reducesto the generalized Rosenau-Kawahara equation when α =
0. Specially, it reduces to the usual Rosenau-Kawaharaequation when a = m = c = λ = ν = , b = , α = ff erential equations, the total accuracy of a particular method is a ff ected not only theorder of accuracy of the method, but also other factors. The conservative property of the method is another factorthat has the same or possibly even more impact on results. For example, one successful and active research is to con-struct structure-preserving schemes (or called symplectic schemes) for the ODE systems (see [62] and the referencestherein). Better solutions can be expected from numerical schemes which have e ff ective conservative properties ratherthan the ones which have nonconservative properties [63]. And Li et al. [64] even pointed out that in some areas, theability to preserve some invariant properties of the original di ff erential equation is a criterion to judge the success ofa numerical simulation.For the initial boundary value problem (11)-(13), we will show that it satisfies a fundamental energy conservationlaw in the next section. In addition, the wave equation is highly nonlinear due to the third nonlinear term in theleft-hand side of the equation (11). When considering the finite di ff erence scheme for the equation (11), the usualCrank-Nicolson scheme will lead to a nonlinear scheme with heavy computation while other standard linearizeddiscretizations for the nonlinear term, e.g., one step Newton’s method or a second-order extrapolation method, will3oss the the energy conservative property. An ideal scheme should have relative less computational cost, can preserveenergy, be unconditionally stable and maintain second-order accuracy.In this paper, a three-level linearly implicit finite di ff erence method for the initial value problem (11)-(13) will bepresented. The fundamental energy conservation is preserved by the presented numerical scheme. The existence anduniqueness of the numerical solution are also proved. Moreover, numerical analysis shows that the method is second-order convergent both in time and space variables, and the method is unconditionally stable. Numerical results confirmwell with the theoretical results.The rest of the paper is organized as follows: Section 2 shows the energy conservation of the initial boundary valueproblem (11)-(13). Section 3 gives the detail description of the three-level linearly implicit finite di ff erence method,the proof for the discrete conservative property, the existence and uniqueness as well as the convergence and stabilityof the numerical solution. Numerical results are shown in section 4. Conclusions are provided in the final section.
2. Conservative property
Equations (11)-(13) satisfy the following energy conservative property.
Theorem 1.
Suppose u ∈ C [ x l , x r ] , then the solution of (11)-(13) satisfies the energy conservation law:E ( t ) = Z x r x l u ( x , t ) + α u x ( x , t ) + λ u xx ( x , t ) dx = Z x r x l u ( x , + α u x ( x , + λ u xx ( x , dx = E (0) , (14) for any t ∈ [0 , T ] , where C [ x l , x r ] is the set of functions which are seventh order continuous di ff erentiable in theinterval [ x l , x r ] and have compact supports inside ( x l , x r ) . Proof.
Multiplying (11) by 2 u and integrating over the interval [ x l , x r ], one get ddt Z x r x l u dx + a Z x r x l uu x dx + b Z x r x l u m + ( u ) x dx + c Z x r x l uu xxx dx − α Z x r x l uu xxt dx + λ Z x r x l uu xxxxt dx − ν Z x r x l uu xxxxx dx = . (15)Using the integration by parts, one can easily obtain Z x r x l uu x dx = Z x r x l udu = (cid:16) u ( x r , t ) − u ( x l , t ) (cid:17) = , Z x r x l u m + ( u ) x dx = Z x r x l u m + du = m + u m + (cid:12)(cid:12)(cid:12) x r x l = , Z x r x l uu xxx dx = uu xx (cid:12)(cid:12)(cid:12) x r x l − Z x r x l u xx du = − Z x r x l u xx u x dx = − Z x r x l u x du x = − u x (cid:12)(cid:12)(cid:12) x r x l = , Z x r x l uu xxt dx = uu xt (cid:12)(cid:12)(cid:12) x r x l − ddt Z x r x l u x dx = − ddt Z x r x l u x dx , Z x r x l uu xxxxt dx = Z x r x l udu xxxt = uu xxxt (cid:12)(cid:12)(cid:12) x r x l − Z x r x l u x u xxxt dx = − u x u xxt (cid:12)(cid:12)(cid:12) x r x l + ddt Z x r x l u xx dx = ddt Z x r x l u xx dx , Z x r x l uu xxxxx dx = uu xxxx (cid:12)(cid:12)(cid:12) x r x l − Z x r x l u xxxx u x dx = − u x u xxx (cid:12)(cid:12)(cid:12) x r x l + Z x r x l u xxx u xx dx = u xx (cid:12)(cid:12)(cid:12) x r x l = , where the boundary conditions (13) are used.Thus, only the first, fifth and sixth term in the left hand side of (15) are nonzero, all other terms are zero. Thisyields, ddt Z x r x l u + α u x + λ u xx dx = . (16)4herefore, E ( t ) = Z x r x l u ( x , t ) + α u x ( x , t ) + λ u xx ( x , t ) dx = Z x r x l u ( x , + α u x ( x , + λ u xx ( x , dx = E (0) , (17)for any t ∈ [0 , T ]. This completes the proof.
3. Numerical method
In this section, we give a complete description of our numerical method for the initial value problem (11)-(13). Wefirst describe our solution domain and its grid. The computational domain is defined as
Ω = { ( x , t ) | x l ≤ x ≤ x r , ≤ t ≤ T } , which is covered by a uniform grid Ω h = { ( x i , t n ) | x i = x l + ih , t n = n τ, i = , · · · , M , n = , · · · , N } , withspacing h = x r − x l M , τ = TN . We denote U ni as the numerical approximation of u ( x i , t n ) and Z h = { U = ( U i ) | U − = U = U = U M − = U M = U M + = , i = − , , , · · · , M − , M , M + } . For convenience, the di ff erence operators, innerproduct and norms are defined as follows:¯ U ni = U n + i + U n − i , ( U ni ) t = U n + i − U n − i τ , ( U ni ) x = U ni + − U ni h , ( U ni ) ¯ x = U ni − U ni − h , ( U ni ) ˆ x = U ni + − U ni − h , ( U n , V n ) = h M − X j = U ni V ni , k U n k = ( U n , U n ) , k U n k ∞ = max ≤ i ≤ M − | U ni | . where U n = ( U n − , · · · , U nM + ) is the numerical solution at time t n = n τ .The essential of our scheme is that the third term in the left-hand side of (11) is rewritten and discretized as bu m u x = bm + u m u x + ( u m + ) x ) ≈ bm + U ni ) m ( ¯ U ni ) ˆ x + (( U ni ) m ¯ U ni ) ˆ x ] , (18)and which is a second-order approximation around ( x i = x l + ih , t n = n τ ). And all other terms of left-hand side of (11)are discretizated by using the standard second-order central di ff erence method around ( x i = x l + ih , t n = n τ ) .The detailed numerical scheme is as follows:( U ni ) t + a ( ¯ U ni ) ˆ x + bm + U ni ) m ( ¯ U ni ) ˆ x + (( U ni ) m ¯ U ni ) ˆ x ] + c ( ¯ U ni ) x ¯ x ˆ x − α ( U ni ) x ¯ xt + λ ( U ni ) xx ¯ x ¯ xt − ν ( ¯ U ni ) xx ¯ x ¯ x ˆ x = , ≤ i ≤ M − , ≤ n ≤ N − . (19)and U i = u ( x i ) , ≤ i ≤ M , (20) U j = U jM = , ( U j ) ˆ x = ( U jM ) ˆ x = , ( U j ) x ¯ x = ( U jM ) x ¯ x = , ≤ j ≤ N . (21)Obviously, the above conditions (21) will give U j = U jM − = U j − = U jM + = ≤ j ≤ N . Thus, U j ∈ Z h , for any 0 ≤ j ≤ N . In addition, we can see that (19) is a three-levellinearly implicit scheme and the coe ffi cient matrix of the linear system (19) is banded. Therefore, the resulting linearsystem of equations can be solved e ffi ciently using a direct linear solver, such as the LU decomposition method.Since the scheme is a three-level method, to start the computation, we need to give the method for computation of U . The U is computed through the following Crank-Nicolson scheme: U i − U i τ + a U i + U i ˆ x + bm + U i + U i m U i + U i ˆ x + U i + U i m U i + U i ˆ x − α U i − U i τ x ¯ x + c U i + U i x ¯ x ˆ x + λ U i − U i τ xx ¯ x ¯ x − ν U i + U i xx ¯ x ¯ x ˆ x = , (22)5here it is a nonlinear scheme and is second-order accurate both in time and space variables.We point out that this discretization (18) for the third term in the left-hand side of (11) is specially designed sothat the numerical scheme (19)-(21) can guarantee the energy conservation (see the proof of theorem 2 in the nextsubsection). Readers can refer to references [29, 40, 52] for the similar kind of treatments for the nonlinear term uu x intheir equations to achieve conservative schemes. In addition, our scheme (19)-(22) is only nonlinear at first time stepwhen computing U , and all other steps are linear. Thus, the computational cost is relatively cheap. In the followingseveral sections of the paper, we will show that our scheme is second-order accurate both in time and space variables,can preserve the energy identity (14), and is unconditionally stable.The following lemmas are well known results, which are essential for existence, uniqueness, convergence, andstability of our numerical solution. Lemma 1 and 2 can be verified through direct computation, lemma 3 is the dis-crete form of the Sobolev’s embedding theorem, which can be found in lemma 1, page 110 of [65]. In the rest partof the paper, unless otherwise indicated, C is the notation referring to a general positive constant, which may havedi ff erent values in di ff erent contexts. Lemma 1.
For any two mesh functions U , V ∈ Z h , one have ( U ˆ x , V ) = − ( U , V ˆ x ) , ( U x , V ) = − ( U , V ¯ x ) , ( U x ¯ x , V ) = − ( U x , V x ) , Furthermore, ( U , U xx ¯ x ¯ x ) = k U x ¯ x k . Lemma 2.
For any mesh function U ∈ Z h , one have ( U ˆ x , U ) = , ( U x ¯ x ˆ x , U ) = , ( U xx ¯ x ¯ x ˆ x , U ) = . Lemma 3. (Discrete Sobolev’s inequality (Lemma 1, page 110 of [65]) For any mesh function U ∈ Z h , one have k U k ∞ ≤ C k U x k . Theorem 2.
Suppose u ∈ C [ x l , x r ] , then the solution of finite di ff erence scheme (19)-(21) satisfies k U n k ∞ ≤ C, k U nx k ∞ ≤ C, for any ≤ n ≤ N, and also satisfies the following discrete energy conservation:E n , k U n + k + k U n k + α k U n + x k + k U nx k + λ k U n + x ¯ x k + k U nx ¯ x k = k U k + k U k + α k U x k + k U x k + λ k U x ¯ x k + k U x ¯ x k , E , (23) for any ≤ n ≤ N − , where E n is the discrete energy at time t = ( n + ) τ . Proof . Taking the conditions U j − = U j = U j = U jM − = U jM = U jM + = ≤ j ≤ N ) into account, and aftercomputing the inner product of equation (19) with ¯ U n , i. e., U n + + U n − , we have k U n + k − k U n − k τ + a ( ¯ U n ˆ x , ¯ U n ) + bm + (cid:16) ( U n ) m ( ¯ U n ) ˆ x + (( U n ) m ¯ U n ) ˆ x , ¯ U n (cid:17) (24) + c ( ¯ U nx ¯ x ˆ x , ¯ U n ) − α ( U nx ¯ xt , ¯ U n ) + λ ( U nxx ¯ x ¯ xt , ¯ U n ) − ν ( ¯ U nxx ¯ x ¯ x ˆ x , ¯ U n ) = . (25)By using lemma 2, we get ( ¯ U n ˆ x , ¯ U n ) = , ( ¯ U nx ¯ x ˆ x , ¯ U n ) = , ( ¯ U nxx ¯ x ¯ x ˆ x , ¯ U n ) = . (26)6oreover, (cid:16)h ( U n ) m (cid:16) ¯ U n (cid:17) ˆ x + (cid:16) ( U n ) m ¯ U n (cid:17) ˆ x i , ¯ U n (cid:17) = M − X i = h ( U ni ) m ( ¯ U ni + − ¯ U ni − ) + ( U ni + ) m ¯ U ni + − ( U ni − ) m ¯ U ni − i ¯ U ni = M − X i = h ( U ni ) m ¯ U ni + ¯ U ni + ( U ni + ) m ¯ U ni + ¯ U ni i − M − X i = h ( U ni − ) m ¯ U ni ¯ U ni − + ( U ni ) m ¯ U ni ¯ U ni − i = . (27)and ( U nxx ¯ x ¯ xt , ¯ U n ) = k U n + x ¯ x k − k U n − x ¯ x k τ , ( U nx ¯ xt , ¯ U n ) = − k U n + x k − k U n − x k τ , (28)where boundary conditions (20) and (21) are used.Thus, k U n + k − k U n − k + α k U n + x k − α k U n − x k + λ k U n + x ¯ x k − λ k U n − x ¯ x k = , (29)for any 1 ≤ n ≤ N −
1. This is equivalent to k U n + k + k U n k + α k U n + x k + k U nx k + λ k U n + x ¯ x k + k U nx ¯ x k = k U n k + k U n − k + α k U nx k + k U n − x k + λ k U nx ¯ x k + k U n − x ¯ x k , (30)for any 1 ≤ n ≤ N −
1. This further yields E n = E , for any 1 ≤ n ≤ N − , (31)which is actually the energy conservation law (23).Multiplying (22) both sides by U i + U i and using the similar techniques as above, one can obtain k U k + α k U x k + λ k U x ¯ x k = k U k + α k U x k + λ k U x ¯ x k . (32)Thus, (31) can be rewritten as E n = k U k + α k U x k + λ k U x ¯ x k . (33)Since u ∈ C [ x l , x r ] and the initial condition (20) are used in the numerical method, the right-side of (33) isbounded. By assumptions, α, λ are positive constants, therefore, k U nx k≤ C , k U nx ¯ x k≤ C , for any 0 ≤ n ≤ N . (34)By using lemma 3, we have k U n k ∞ ≤ C .In addition, through direct computation, one can verify that k U nxx k = k U nx ¯ x k , for any 0 ≤ n ≤ N . (35)Thus, k U nxx k≤ C , for any 0 ≤ n ≤ N . (36)Again by using lemma 3, we have k U nx k ∞ ≤ C . This completes the proof.7 .3. Existence and uniqueness Theorem 3.
The finite di ff erence scheme (19)-(21) has a unique solution. Proof . To prove the theorem, we proceed by the mathematical induction. Suppose U , · · · , U n (1 ≤ n ≤ N − U n + . Assume that U n + , , U n + , are two solutions of (19)and let W n + = U n + , − U n + , , then it is easy to verify that W n + satisfies the following equation:12 τ W n + i + a W n + i ) ˆ x + b m + h ( U ni ) m ( W n + i ) ˆ x + (( U ni ) m W n + i ) ˆ x i + c W n + i ) x ¯ x ˆ x − α τ ( W n + i ) x ¯ x + λ τ ( W n + i ) xx ¯ x ¯ x − ν W n + i ) xx ¯ x ¯ x ˆ x = . (37)Taking the inner product of (37) with W n + , we have12 τ k W n + k + α τ k W n + x k + λ τ k W n + x ¯ x k = , (38)where ( W n + x ¯ x , W n + ) = − ( W n + x , W n + x ) , ( W n + xx ¯ x ¯ x , W n + ) = k W n + x ¯ x k , ( W n + xx ¯ x ¯ x ˆ x , W n + ) = , ( W n + x ¯ x ˆ x , W n + ) = , ( W n + x , W n + ) = , (cid:16)h ( U n ) m (cid:16) W n + (cid:17) ˆ x + (cid:16) ( U n ) m W n + (cid:17) ˆ x i , W n + (cid:17) = , (39)are used. The first five identities of (39) are directly from lemma 1 and 2, and the last one can be obtained as follows: (cid:16)h ( U n ) m (cid:16) W n + (cid:17) ˆ x + (cid:16) ( U n ) m W n + (cid:17) ˆ x i , W n + (cid:17) = M − X i = h ( U ni ) m ( W n + i + − W n + i − ) + ( U ni + ) m W n + i + − ( U ni − ) m W n + i − i W n + i = M − X i = h ( U ni ) m W n + i + W n + i + ( U ni + ) m W n + i + W n + i i − M − X i = h ( U ni ) m W n + i − W n + i + ( U ni − ) m W n + i − W n + i i = . (40)From (38) and the definition of the k · k -norm, one can see that (38) has only a trivial solution. Thus, (19)determines U n + uniquely. This completes the proof. Let u ( x , t ) be the solution of problem (11)-(13), U ni be the solution of the numerical schemes (19)-(21), and u ni = u ( x i , t n ) , e ni = u ni − U ni , then the truncation error of the scheme (19)-(21) can be obtained as follows: r ni = (cid:0) e ni (cid:1) t + a (¯ e ni ) ˆ x + bm + h ( u ni ) m ( ¯ u ni ) ˆ x + (cid:0)(cid:0) u ni (cid:1) m ¯ u ni (cid:1) ˆ x − (cid:0) U ni (cid:1) m (cid:16) ¯ U ni (cid:17) ˆ x − (cid:16)(cid:0) U ni (cid:1) m ¯ U ni (cid:17) ˆ x i + c (cid:0) ¯ e ni (cid:1) x ¯ x ˆ x − α ( e ni ) x ¯ xt + λ ( e ni ) xx ¯ x ¯ xt − ν (cid:0) ¯ e ni (cid:1) xx ¯ x ¯ x ˆ x , (41)where ¯ e n = e n + + e n − , 2 ≤ i ≤ M − ≤ n ≤ N − x i = x l + ih , t n = n τ ), by Taylor expansion, it can be easily obtained that r ni = O ( τ + h ) if h , τ → u ( x , t ) ∈ C , , where C , is the set of functions which are seventh order continuous di ff erentiable in space and thirdorder continuous di ff erentiable in time. This following lemma is a well known result.8 emma 4. (Discrete Gronwall’s inequality). Suppose that w ( k ) and ρ ( k ) are nonnegative functions while ρ ( k ) is anon-decreasing function. If w ( k ) ≤ ρ ( k ) + C τ k − X l = w ( l ) , ∀ k , then w ( k ) ≤ ρ ( k ) e C τ k , ∀ k . Theorem 4.
Suppose u ∈ C [ x l , x r ] , and u ( x , t ) ∈ C , , then the numerical solution U n of the finite di ff erence scheme(19)-(21) converges to the solution of the problem (11)-(13) in the sense of k · k ∞ , and the convergence rate isO ( τ + h ) , i.e., k u n − U n k ∞ ≤ C ( τ + h ) , for any ≤ n ≤ N . (42) Proof . Taking the inner product of (41) with 2¯ e n , we have k e n + k − k e n − k + α ( k e n + x k − k e n − x k ) + λ ( k e n + x ¯ x k − k e n − x ¯ x k ) = τ h ( r n , e n ) − a (cid:16) ¯ e n ˆ x , e n (cid:17) − c ((¯ e n ) x ¯ x ˆ x , e n ) + ν ((¯ e n ) xx ¯ x ¯ x ˆ x , e n ) − ( Q , e n ) − ( R , e n ) i , (43)where Q = bm + h ( u n ) m (¯ u n ) ˆ x − ( U n ) m (cid:16) ¯ U n (cid:17) ˆ x i , R = bm + h(cid:0) ( u n ) m ¯ u n (cid:1) ˆ x − (cid:16) ( U n ) m ¯ U n (cid:17) ˆ x i . (44)By using lemma 2, we obtain (cid:16) ¯ e n ˆ x , e n (cid:17) = , ((¯ e n ) x ¯ x ˆ x , e n ) = , ((¯ e n ) xx ¯ x ¯ x ˆ x , e n ) = . (45)From the notations introduced at the beginning of section 3, for any 0 ≤ n ≤ N , we have the following inequality k e n ˆ x k = M − X j = (cid:18) e j + − e j − h (cid:19) h = M − X j = (cid:18) e j + − e j h + e j − e j − h (cid:19) h ≤ M − X j = "(cid:18) e j + − e j h (cid:19) + (cid:18) e j − e j − h (cid:19) h = k e nx k . Thus, for any 0 ≤ n ≤ N , we have k e n ˆ x k≤k e nx k . (46)In addition, we have | ( Q , e n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bhm + M − X j = h ( u nj ) m ( ¯ u nj ) ˆ x − ( U nj ) m ( ¯ U nj ) ˆ x i ¯ e nj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bhm + b M − X j = h ( u nj ) m ( ¯ u nj ) ˆ x − ( u nj ) m ( ¯ U nj ) ˆ x + ( u nj ) m ( ¯ U nj ) ˆ x − ( U nj ) m ( ¯ U nj ) ˆ x i ¯ e nj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bhm + M − X j = ( u nj ) m (¯ e nj ) ˆ x ¯ e nj + bhm + M − X j = h(cid:16) u nj (cid:17) m − ( U nj ) m i ( ¯ U nj ) ˆ x ¯ e nj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bhm + M − X j = ( u nj ) m (¯ e nj ) ˆ x ¯ e nj + bhm + M − X j = m − X k = ( u nj ) m − − k ( U nj ) k e nj ( ¯ U nj ) ˆ x ¯ e nj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch M − X j = h | (¯ e nj ) ˆ x | + | e nj | i | ¯ e nj |≤ C ( k e n + x k + k e n − x k + k e n + k + k e n k + k e n − k ) , ≤ C ( k e n + x k + k e n − x k + k e n + k + k e n k + k e n − k ) , (47)9here theorem 2, Cauchy-Schwarz inequality and inequality (46) are used.Similarly, we have | ( R , e n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bhm + M − X j = h(cid:16) ( u nj ) m ¯ u nj (cid:17) ˆ x − (cid:16) ( U nj ) m ¯ U nj (cid:17) ˆ x i ¯ e nj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bhm + M − X j = h(cid:16) ( u nj ) m ¯ u nj (cid:17) ˆ x − (cid:16) ( u nj ) m ¯ U nj (cid:17) ˆ x + (cid:16) ( u nj ) m ¯ U nj (cid:17) ˆ x − (cid:16) ( U nj ) m ¯ U nj (cid:17) ˆ x i ¯ e nj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bhm + M − X j = (cid:16) ( u nj ) m ¯ e nj (cid:17) ˆ x + m − X k = ( u nj ) m − − k ( U nj ) k e nj ¯ U nj ˆ x ¯ e nj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − bhm + M − X j = ( u nj ) m ¯ e nj + m − X k = ( u nj ) m − − k ( U nj ) k ¯ U nj e nj (¯ e nj ) ˆ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch M − X j = h | ¯ e nj | + | e nj | i | (¯ e nj ) ˆ x |≤ C ( k e n + x k + k e n − x k + k e n + k + k e n k + k e n − k ) , ≤ C ( k e n + x k + k e n − x k + k e n + k + k e n k + k e n − k ) , (48)where theorem 2, Cauchy-Schwarz inequality and inequality (46) are used again.Furthermore, we have( r n , e n ) ≤ k r n kk ¯ e n k≤k r n k + k ¯ e n k ≤k r n k + (cid:16) k e n + k + k e n − k (cid:17) , (49)where Cauchy-Schwarz inequality are used.Substituting (47)-(49) into (43), we get k e n + k − k e n − k + α ( k e n + x k − k e n − x k ) + λ ( k e n + x ¯ x k − k e n − x ¯ x k ) ≤ C τ ( k e n + k + k e n k + k e n − k + k e n + x k + k e n − x k ) + τ k r n k . (50)Since α, λ are positive constants, it is easy to check that k e n + k − k e n − k + α ( k e n + x k − k e n − x k ) + λ ( k e n + x ¯ x k − k e n − x ¯ x k ) ≤ C τ ( k e n + x k + k e nx k + k e n + k + k e n k + k e n − k ) + τ k r n k ≤ C ′ τ ( k e n + k + k e n k + k e n − k + α k e n + x k + α k e nx k ) + τ k r n k , where C ′ = max( C α , C ) and C is the positive constant in the above inequality.Replacing C ′ in the above inequality by the general positive constant notation C , we have k e n + k − k e n − k + α ( k e n + x k − k e n − x k ) + λ ( k e n + x ¯ x k − k e n − x ¯ x k ) ≤ C τ ( k e n + k + k e n k + k e n − k + α k e n + x k + α k e nx k ) + τ k r n k ≤ C τ ( k e n + k + k e n k + k e n − k + α k e n + x k + α k e nx k + α k e n − x k + λ k e n + x ¯ x k + λ k e nx ¯ x k + λ k e n − x ¯ x k ) + τ k r n k . (51)Let D n = k e n k + α k e nx k + λ k e nx ¯ x k + k e n − k + α k e n − x k + λ k e n − x ¯ x k , D n + − D n ) ≤ C τ ( D n + + D n ) + τ k r n k , which is equivalent to (1 − C τ )( D n + − D n ) ≤ C τ D n + τ k r n k . (52)If τ , which is su ffi ciently small, satisfies τ < C ( C is the positive constant in the inequality (52)), then 1 − C τ > D n + − D n ≤ C − C τ τ D n + τ − C τ k r n k ≤ C τ D n + τ k r n k , ≤ C ′′ ( τ D n + τ k r n k ) , where C ′′ = max(3 C ,
3) and we have used − C τ < τ < C .Replacing C ′′ in the above inequality by the general positive constant notation C , we have D n + − D n ≤ C τ D n + C τ k r n k . (53)Summing (53) from 1 to n −
1, we get D n ≤ D + C τ n − X l = D l + C τ n − X l = k r l k , (54)where τ n − X l = k r l k ≤ n τ max ≤ l ≤ n − k r l k ≤ T · O ( τ + h ) . (55)Since e i = U , we have D = O ( τ + h ) followed by asimple analysis for the scheme (22). Therefore D n ≤ O ( τ + h ) + C τ n − X l = D l . (56)Using lemma 4, we obtain D n ≤ O ( τ + h ) . (57)Thus, k e n k≤ O ( τ + h ) . (58)By using lemma 3, we have k e n k ∞ ≤ O ( τ + h ) , (59)i.e., k u n − U n k ∞ ≤ C ( τ + h ) . (60)This completes the proof. Theorem 5.
Suppose u ∈ C [ x l , x r ] , then the solution U n of the finite di ff erence scheme (19)-(21) is unconditionallystable with the k · k ∞ norm. The proof of this theorem is similar as the above theorem.11 . Numerical results
In order to preform numerical accuracy test of the method proposed in this paper, one needs to know the exactsolitary solutions beforehand. Exact solitary solutions of the equation (11) were previously obtained by [61] using thesech ansatz method. In this paper, we provide a di ff erent derivation through the sine-cosine method, which is listed inthe appendix A. Example 1.
We present the numerical results for the case m = , a = , b = , c = , α = , λ = , ν =
1. From[61] and appendix A, one can obtain the exact solitary solution as follows: u ( x , t ) = √ − √ q √ −
29 sech q √ − x − − √ √ − t , (61)and the initial condition is set as u ( x , = √ − √ q √ −
29 sech q √ − x . (62)We first carry out the numerical convergence studies. For the spatial convergence, we set τ = .
005 as the fixedtime step and use 4 di ff erent spatial meshes: h = . , . , . , .
1, where τ is su ffi cient small such that the temporalerror is negligible comparing to the spatial error (Here the time step is τ = .
005 while the smallest spatial size is h = .
1, thus h >> τ , therefore, the dominant errors are the spatial errors). The final time T is set to be 10, and x l = − , x r = h = .
005 as the fixed spatial mesh and use 4 di ff erent temporal meshes: τ = . , . , . , .
1, where h issu ffi cient small such that the spatial error is negligible comparing to the temporal error (Here the smallest time step is τ = . h = . h << τ , therefore, the dominant errors are the temporal errors). Thefinal time T is set to be 10, and x l = − , x r = T = , x l = − , x r = , h = . , τ = . E n atseveral time stages. We can see that E n is conserved exactly (up to 8 decimals) during the time evolution of the solitarywave. Example 2.
We present the numerical results for the case m = , a = , b = , c = , α = , λ = , ν =
1. From[61] and appendix A, one can obtain the exact solitary solution as follows: u ( x , t ) = √ − √ − sech q √ − x − − √ √ − t , (63)and the initial condition is set as u ( x , = √ − √ − sech q √ − x . (64)Again, we carry out the spatial and temporal convergence. Table 4 and 5 give the errors between numericalsolutions and exact solutions for spatial and temporal convergence, respectively. Once again, we can see that themethod is second-order convergent both in time and space variables.Additionally, table 6 provides the quantities of E n with T = , x l = − , x r = , h = . , τ = .
1. Once again,we can see that E n is conserved and the method can be well used to study the solitary wave at long time.12 . Conclusions In this paper, a three-level linearly implicit finite di ff erence method for the initial value problem of the generalizedRosenau-Kawahara-RLW equation is developed. The fundamental energy conservative property is preserved by thecurrent numerical scheme. The existence and uniqueness of the numerical solution are also proved. The method isshown to be second-order convergent both in time and space variables, and it is unconditionally stable. Moreover,exact solitary solutions are derived through sine-cosine method which is used for the numerical tests. Numericalresults confirm well with the theoretical results. Acknowledgement
Dongdong He was supported by the Program for Young Excellent Talents at Tongji University (No. 2013KJ012),the Natural Science Foundation of China (No. 11402174) and the Scientific Research Foundation for the ReturnedOverseas Chinese Scholars, State Education Ministry. Kejia Pan was supported by the Natural Science Foundation ofChina (Grant Nos. 41474103 and 41204082), the Natural Science Foundation of Hunan Province of China (Grant No.2015JJ3148) and the Mathematics and Interdisciplinary Sciences Project of Central South University.
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10 for example 1. h k e k rate k e k ∞ rate0 . . . . able 2: Temporal mesh refinement analysis with h = . , T =
10 for example 1. τ k e k rate k e k ∞ rate0.8 1.523 - 6.060e-1 -0.4 3.812e-1 1.999 1.540e-1 1.9760.2 9.711e-2 1.973 3.910e-2 1.9770.1 2.442e-2 1.992 9.820e-3 1.99317 able 3: Invariant of E n for example 1. t E n able 4: Spatial mesh refinement analysis with τ = . , T =
10 for example 2. h k e k rate k e k ∞ rate0 . . . . able 5: Temporal mesh refinement analysis with h = . , T =
10 for example 2. τ k e k rate k e k ∞ rate0.8 4.582-1 - 2.029e-1 -0.4 8.633e-2 2.408 3.843e-2 2.4010.2 2.124e-2 2.023 9.447e-3 2.0240.1 5.263e-3 2.012 2.330e-3 2.01920 able 6: Invariant of E n for example 2. t E n ppendix A. Exact solitary solutions The sine-cosine method uses the sine or cosine function as the wave function to seek the traveling wave solution ofa time dependent partial di ff erential equation, which has the advantage of reducing the nonlinear problem to a systemof algebraic equations that can be easily solved by using a symbolic computation system such as Mathematica orMaple [3, 38, 40, 66].For equation (11), one can obtain the exact solitary solutions by using the sine-cosine method. Firstly, we seek thefollowing travelling wave solution [3, 38, 40, 66]: u ( x , t ) = ˆ u ( ξ ) , ξ = x − vt , (A.1)where v is referred as the wave velocity which is a constant to be determined later.Under the transformation of (A.1) and integrating once, (11) can be reduced into:( a − v ) ˆ u + bm + u m + + ( v α + c ) ˆ u ξξ − ( λ v + ν ) ˆ u ξξξξ = . (A.2)Secondly, by using the sine-cosine method, the solutions of the above reduced ODE equation can be expressed in thefollowing form [38, 40, 66]: ˆ u ( ξ ) = ( A cos η ( B ξ ) , if | ξ | < π B , , otherwise , (A.3)or in the form ˆ u ( ξ ) = ( A sin η ( B ξ ) , if | ξ | < π B , , otherwise , (A.4)where A , B , η are parameters to be determined.Using (A.3), one have ˆ u ξξ = AB η ( η −
1) cos η − ( B ξ ) − AB η cos η ( B ξ ) , (A.5)and ˆ u ξξξξ = AB η ( η − η − η −
3) cos η − ( B ξ ) − AB η ( η − η − η +
2) cos η − ( B ξ ) + AB η cos η ( B ξ ) . (A.6)Substituting (A.3), (A.5) and (A.6) into (A.2), one obtain bA m + m + η ( m + ( B ξ ) − AB η ( η − η − η − λ v + ν ) cos η − ( B ξ ) + ( AB ( v α + c ) η ( η − + AB η ( η − η − η + λ v + ν )) cos η − ( B ξ ) + (cid:16) ( a − v ) A − AB ( v α + c ) η − AB η ( λ v + ν ) (cid:17) cos η ( B ξ ) = . (A.7)Balancing cos η ( m + ( B ξ ) and cos η − ( B ξ ), and setting each coe ffi cients of cos j ( B ξ ) ( j = η, η − , η −
4) to be zero, onecan obtain a set of equations for η, v , A , B as follows: η ( m + = η − , (A.8) bA m + m + − AB η ( η − η − η − λ v + ν ) = , (A.9) AB ( v α + c ) η ( η − + AB η ( η − η − η + λ v + ν ) = , (A.10)( a − v ) A − AB ( v α + c ) η − AB η ( λ v + ν ) = . (A.11)From (A.8), one can find that η = − m . (A.12)22rom (A.10) and (A.11), one obtain that B = ( λ a + ν )( η − η + ± p ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − ( λ c − να ) η ( η − . (A.13)Once B is obtained, one can get that v = − ν B ( η − η + + c λ B ( η − η + + α , (A.14) A = " m + b B η ( η − η − η − λ v + ν ) m . (A.15)The classification for the above solution is discussed as follows:Case 1: ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − > ( λ a + ν )( η − η + + √ ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − ( λ c − να ) η ( η − >
0. We obtain a periodic solution as follows: u ( x , t ) = ˆ u ( ξ ) = A cos η ( B ( x − vt )) , (A.16)where B is given by B = s ( λ a + ν )( η − η + + p ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − ( λ c − να ) η ( η − , (A.17)and η, v , A are given by (A.12), (A.14) and (A.15), respectively.(2): ( λ a + ν )( η − η + + √ ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − ( λ c − να ) η ( η − <
0. We obtain a solitary solution as follows: u ( x , t ) = ˆ u ( ξ ) = A sech η ( B ( x − vt )) , (A.18)where B is given by B = s − ( λ a + ν )( η − η + − p ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − ( λ c − να ) η ( η − , (A.19)and η, v , A are given by (A.12), (A.14) and (A.15), respectively.(3): ( λ a + ν )( η − η + − √ ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − ( λ c − να ) η ( η − >
0. Similar periodic solution as (A.16) can be obtained.(4): ( λ a + ν )( η − η + − √ ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − ( λ c − να ) η ( η − <
0. Similar solitary solution as (A.18) can be obtained.Case 2: ( λ a + ν ) ( η − η + + ( λ c − να )( α a + c ) η ( η − < A , B , v are complex numbers with nonzero real parts. After substituting the expressionsof A , B , v , η into equation (A.3), uu