A new conservative finite difference scheme for the Rosenau equation

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Abstract

A conservative difference scheme is given for the KdV like Rosenau equation. The unique solvability of numerical solutions is shown. A priori bound, convergence and stability of the difference scheme are proved.

Introduction

In recent years, a vast amount of work and computation has been devoted to the initial value problem for the Korteweg-de Vries (KdV) equation, which describes the vibrations of a unidimensional anharmonic lattice associated with the birth of the soliton. To overcome the shortcomings of KdV equation, Rosenau [1] has developed a model called the Rosenau equation, to describe the dynamics of dense discrete systems.

The Rosenau equation is given byut+uxxxxt+ux+uux=0,xΩ,t(0,T]with the boundary conditionsu(x,t)=uxx(x,t)=0,xΩ,t(0,T],and an initial conditionu(x,0)=u0(x),xΩ¯,where Ω=(0,1) and 0<T<+.

The following conservation law is known ast[0,T],E(t)=u(·,t)L22+uxx(·,t)L22=u0L22+u0xxL22=E(0).

The theoretical results on existence, uniqueness and regularity of solutions to (1.1a), (1.1b), (1.1c) have been investigated by Park [2]. Using Faedo–Galerkin method, Chung and Pani in [3] have established well-posedness and some regularity results of the above problem. The numerical approximation to the problem (1.1a), (1.1b), (1.1c) in one dimensional space has been developed by Chung and Ha [4] using finite element Galerkin method, where they obtained optimal H2-estimates and suboptimal order error estimates in L2 and L-norms.

However, Chung and Pani [3] have considered the Rosenau equation in several space variables and have derived optimal error estimates in L2 and quasi-optimal estimates in L-norm. For both these articles, finite element spaces consisting of C1-piecewise polynomials of degree r2 are used. Subsequently, in [5] using the splitting method, C0-piecewise linear spaces are employed in conjunction with the lumped mass finite element method and a priori bounds are derived. Manickam et al. [6] have developed the numerical solution of (1.1a), (1.1b), (1.1c) using an orthogonal cubic spline collocation method. Convergence of the semi discrete solution of the Rosenau equation and the error analysis of the modified Galerkin–Crank–Nicolson method has been studied in [7], [8].

In spite of computational convenience of finite difference methods, there are few studies on finite difference methods compared to large amount of studies on finite element approximate solutions for (1.1a), (1.1b), (1.1c) because of uneasy control of nonlinear term uux. In [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] the conservation finite difference schemes were used for a system of the generalized nonlinear Schrödinger equation, regularized long wave equation, sine-Gordon equation, Klein–Gordon equation, Zakharov equation and Cahn–Hilliard equation, respectively.

In [27], Chung has proposed a nonlinear conservative scheme for the initial boundary value problem (1.1a), (1.1b), (1.1c). In this paper, we propose a three-level finite difference scheme for the Rosenau equation (1.1a), (1.1b), (1.1c) which is conservative and unconditionally stable, and we prove error estimates of second-order. The remainder of this article is organized as follows; in Section 2 we describe a finite difference for the Rosenau equation. In Section 3, we show that the scheme is energy conserving. In Section 4, we prove that the difference scheme is uniquely solvable. In Section 5, we derive a priori error estimates for numerical solution. In Section 6, we prove the convergence and stability for our difference scheme.

Section snippets

Finite difference scheme

Let h=1M be the uniform step size in the spatial direction for a positive integer M and k=TN denotes the uniform step size in the temporal direction for a positive integer N. Denote Vin=V(xi,tn) for tn=nk, n=0,1,,N and Sh={V=(Vi)/V0=VM=0,i=0,1,,M}.

We define the difference operators, for a function VSh ash+Vin=Vi+1n-Vinh,h-Vin=Vin-Vi-1nh,hVin=Vi+1n-Vi-1n2h,ΔhVin=h+(h-Vin),Δh2Vin=Δh(ΔhVin).Further, we define operators V¯n and tVn, respectively, asV¯in=Vin+1+Vin-12,tVin=Vin+1-Vin-12k.We

Discrete conservative law

To obtain conservative law, we introduce the following lemma:

Lemma 1

For UnSh, we have(1)(hU¯n,U¯n)h=0,(2)(ψ(Un,U¯n),U¯n)h=0.

Proof

(1) In view of difference properties and the boundary conditions (2.1d), we obtain(hU¯n,U¯n)h=hi=1M-1(hU¯in)U¯in=12i=1M-1(U¯i+1n-U¯i-1n)U¯in=12i=1M-1U¯i+1nU¯in-i=1MU¯i-1nU¯in=0.(2) For UnSh, we have(ψ(Un,U¯n),U¯n)h=124i=1M-1[Uin(Ui+1n+1+Ui+1n-1-Ui-1n+1-Ui-1n-1)+Ui+1n(Ui+1n+1+Ui+1n-1)-Ui-1n(Ui-1n+1+Ui-1n-1)](Uin+1+Uin-1).Noting thati=1M-1[Uin(Ui+1n+1+Ui+1n-1)+Ui+1n(Ui+1n

Solvability

Below, we are going to prove the solvability of the finite difference scheme (2.1a), (2.1b), (2.1c), (2.1d).

Theorem 2

The difference scheme (2.1a), (2.1b), (2.1c), (2.1d) is uniquely solvable.

Proof

It is obvious that U0 and U1 are uniquely determined by (2.1b), (2.1c). Now suppose U0,U1,,Un(1nN-1) be solved uniquely. Consider the equation of (2.1a) for Un+112kUin+1+12kh2Uin+1+1-θ2hUin+1+16[UinhUin+1+h(UinUin+1)]=0.Computing the inner product of Eq. (4.1) with 2Un+1, and using (3.1), we obtain1kUn+1h2+1

Some priori estimates for the difference solution

The following Lemma can be verified by summation by parts and using minimum eigenvalue of symmetric matrix. For a proof, see Agarwall [28].

Lemma 2

For function V defined on Sh, the following identity and inequality hold:1.-(hV,V)h=|V|1,h2.2.The discrete Poincare´inequality:2sin(πh2)hVh|V|1,h.3.2sin(πh2)h|V|1,h|V|2,h.

Next, we shall use the following Lemma [29].

Lemma 3 Discrete Sobolev’s inequality

Suppose that {Ui} is mesh functions. Given ε>0, there exists a constant C dependent on ε such thatU,hε|U|1,h+CUh.

Lemma 4

Suppose that u0 is

Convergence

Define the net function uin=u(xi,tn) and let Ein=uin-Uin andσ=ψ(un,u¯n)-ψ(Un,U¯n),E¯nh.

Lemma 5

Assume that the solution u(x,t) of (1.1a), (1.1b), (1.1c) is sufficiently regular, there exists a positive constant C independent of h and k such thatσC[En+1h2+Enh2+En-1h2+|En+1|2,h2+|En-1|2,h2].

Proof

It follows from (2.2), (5.13) thatσ=13unhu¯n-UnhU¯n+h(unu¯n)-h(UnU¯n),E¯nh=13hi=1M-1uinhE¯in+EinhU¯in+h(uinE¯in)+h(EinU¯in)E¯in=13hi=1M-1uinhE¯in+EinhU¯inE¯in-uinE¯in+EinU¯inhE¯in.Using Theorem 3,

Conclusion

In this paper, we have proposed a new conservative finite difference scheme for the Rosenau equation, which has a wide range of applications in physics.

A second-order error estimates as well as conservation of energy, and stability of the finite difference approximate solutions were discussed in detail.

Acknowledgements

The authors thank the referees for their valuable comments which improved this article.

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