Conservative Linear Difference Scheme for Rosenau-KdV Equation

A conservative three-level linear finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-KdV equation is proposed.The difference scheme simulates two conservative quantities of the problemwell.The existence and uniqueness of the difference solution are proved. It is shown that the finite difference scheme is of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.


Introduction
KdV equation has been used in very wide applications and undergone research which can be used to describe wave propagation and spread interaction as follows [1][2][3][4]: In the study of the dynamics of dense discrete systems, the case of wave-wave and wave-wall interactions cannot be described using the well-known KdV equation.To overcome this shortcoming of the KdV equation, Rosenau [5,6] proposed the so-called Rosenau equation: The existence and the uniqueness of the solution for (2) were proved by Park [7], but it is difficult to find the analytical solution for (2).Since then, much work has been done on the numerical method for (2) ( [8][9][10][11][12][13] and also the references therein).On the other hand, for the further consideration of the nonlinear wave, the viscous term +  needs to be included [14]   +   +   +   +   = 0. ( This equation is usually called the Rosenau-KdV equation. Zuo [14] discussed the solitary wave solutions and periodic solutions for (2).Recently, [15][16][17] discussed the solitary solutions for the generalized Rosenau-KdV equation with usual power law nonlinearity.In [15,16], the authors also gave the two invariants for the generalized Rosenau-KdV equation.In particular, [16] not only derived the singular 1-solition solution by the ansatz method but also used perturbation theory to obtain the adiabatic parameter dynamics of the water waves.In [17], The ansatz method is applied to obtain the topological soliton solution of the generalized Rosenau-KdV equation.The   / method as well as the exp-function method are also applied to extract a few more solutions to this equation.But the numerical method to the initialboundary value problem of Rosenau-KdV equation has not been studied till now.In this paper, we propose a conservative three-level finite difference scheme for the Rosenau-KdV equation (3) with the boundary conditions and an initial condition
It is known the conservative scheme is better than the nonconservative ones.The nonconservative scheme may easily show nonlinear blow up.A lot of numerical experiments show that the conservative scheme can possesses some invariant properties of the original differential equation [18][19][20][21][22][23][24][25][26][27][28][29].The conservative scheme is more suitable for longtime calculations.In [19], Li and Vu-Quoc said ". . . 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." In this paper, we propose a three-level linear finite difference scheme for the Rosenau-KdV equation ( 3)- (5).The difference scheme is conservative which simulates conservative properties ( 6) and (7) at the same time.
The rest of this paper is organized as follows.In Section 2, we propose a three-level linear finite difference scheme for the Rosenau-KdV equation and discuss the discrete conservative properties.In Section 3, we show that the scheme is uniquely solvable.Then, in Section 4, we prove that the finite difference scheme is of second-order convergence, unconditionally stable.Finally, some numerical tests are given in Section 5 to verify our theoretical analysis.

Lemma 1. It follows from summation by parts that for any two mesh functions
Then one has Furthermore, if The difference scheme ( 10)-( 13) simulates two conservative properties of the problems ( 6) and (7) as follows.
Proof.Use mathematical induction to prove it.It is obvious that  0 is uniquely determined by the initial condition (12).

Convergence and Stability
Let V(, ) be the solution of problem (3)-( 5), V   = V(  ,   ), then the truncation error of the difference scheme (10)-( 13) is as follows:  5,3 , then the solution   of the difference scheme (10)-( 13) converges to the solution V(, ) of the problem (3)-( 5) with order ( Proof.Subtracting ( 10) from ( 29) and letting    = V   −    , we have where Computing the inner product of (39) with 2  , we obtain Similar to the proof of ( 21), we have Then, (40) can be rewritten as follows: According to the Schwarz inequality, we obtain Noting that (48) Then, (48) can be rewritten as follows: which yields If  is sufficiently small, which satisfies 1 −  > 0, then Summing up (51) from 1 to , we have First, we can get  1 in order ( 2 + ℎ 2 ) that satisfies  0 = ( 2 + ℎ 2 ) 2 by two-level - scheme.Since then we obtain From Lemma 6 we get Finally, we can similarly prove results as follows.

Numerical Simulations
Since the three-level implicit finite difference scheme cannot start by itself, we need to select other two-level schemes (such as the - Scheme) to get  1 .Then, be reusing initial value  0 , we can work out  2 ,  (59)  In Table 1, we give the error at various time steps.Using the method in [30,31], we verified the second convergence of the difference scheme in Table 2. Numerical simulations on two conservation invariants   and   are given in Table 3.The wave graph comparison of (, ) between  = ℎ = 0.1 and  = ℎ = 0.025 at various times is given in Figures 1  and 2.
Numerical simulations show that the finite difference scheme is efficient.

Table 1 :
The error at various time steps.

Table 3 :
Numerical simulations on the two conservation invariants   and   .