A High-Accuracy Linear Conservative Difference Scheme for Rosenau-RLW Equation

We study the initial-boundary value problem for Rosenau-RLW equation. We propose a three-level linear finite difference scheme, which has the theoretical accuracy of O(τ2 + h). The scheme simulates two conservative properties of original problem well. The existence, uniqueness of difference solution, and a priori estimates in infinite norm are obtained. Furthermore, we analyze the convergence and stability of the scheme by energy method. At last, numerical experiments demonstrate the theoretical results.


Introduction
In the study of the dynamics of compact discrete systems, 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 Rosenau equation in [1,2]: Rosenau equation ( 1) 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.The existence and uniqueness of solution to (1) were proved by Park in [3].Rosenau equation is also regarded as the transformation of the Regularized Long Wave (RLW) equation (see [4]): which is usually used to simulate the long wave in nonlinear emanative medium.RLW equation plays important role in the study of nonlinear diffusion wave because it could model lots of physical phenomena.As RLW equation and KdV equation have the same approximative order when they are used to describe motivations, RLW equation could simulate almost all of the applications of KdV equation [5].Therefore, there are many works about Rosenau equation (1) and RLW equation (2) (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]).
It is easy to verify that problem (3)-( 5) satisfies the following conservative laws [27]: where (0) and (0) are both constants only depending on initial data.Li and Vu-Quoc pointed in [28] 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.It is said in [29] that conservative difference scheme can simulate the conservative law of initial problem well and it could avoid the nonlinear blowup.Therefore, constructing conservative difference scheme is an important and significant job.To our knowledge, the theoretic accuracy of the existing difference scheme for Rosenau-RLW equation (see [24,25,27]) is ( 2 + ℎ 2 ).Particularly, in [27], the authors proposed a three-level linear conservative difference scheme for problem (3)-( 5), whose theoretic accuracy is ( 2 + ℎ 2 ).One does not need iteration when solving the equation numerically using this scheme.Henceforth, it could save some computing time.Using Richardson extrapolation idea, we will propose a three-level linear difference scheme which has the theoretic accuracy of ( 2 + ℎ 4 ) without refined mesh in this paper.Our scheme simulates the two conservative laws ( 7) and ( 8) well.And we will study the a priori estimate, existence, and uniqueness of the difference solution.Furthermore, we shall analyze the convergence and stability.
The rest of this paper is organized as follows.We propose the conservative difference scheme in Section 2 and prove the existence and uniqueness of solution to difference scheme by mathematical induction in Section 3. Section 4 is devoted to the prior estimate, convergence, and stability of the difference scheme.In Section 5, we verify our theoretical analysis by numerical experiments.
Taking an inner product of (12) and 2  (i.e.,  +1 +  −1 ), we could obtain from boundary condition (15) and summation by parts [30] where On the other hand, Taking ( 23)-( 25) into ( 21), we have From the definition of   , we know that (17) could be obtained by deducing the above equality from 0 to  − 1.
Proof.Subtracting ( 12)-( 15) from (32) and letting    = V   −   , we have Taking the inner product of (43) and  1 , and using boundary condition (44), we obtain Noticing that we can obtain that Similar to (23), we have By using Lemma 4, Theorem 5, Lemma 1, and the Cauchy-Schwarz inequality, we can get The discrete Gronwall inequality (see [30]) implies that from the discrete Sobolev inequality.

Numerical Simulations
As the difference scheme ( 12)-( 15) is a linear system about  +1  , it does not need iteration when we solve it numerically.
For some different values of  and ℎ, we list errors at several times in Table 1 and verify the accuracy of the difference scheme in Table 2.The numerical simulation of two conservative quantities (7) and ( 8) is listed in Table 3.
The stability and convergence of the scheme are verified by these numerical experiments.And it shows that our proposed algorithm is effective and reliable.