# Numerical Simulation for General Rosenau-RLW Equation: An

Authors: Xintian Pan; Luming Zhang
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Numerical solutions for the general Rosenau-RLW equation are considered and
an energy conservative linearized finite difference scheme is proposed. Existence of the solutions
for the difference scheme has been shown. Stability, convergence, and a priori error estimate of the
scheme are proved using energy method. Numerical results demonstrate that the scheme is efficient
and reliable.


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## Body

## 1. Introduction

In this paper, we examine the use of the finite difference method for the general Rosenau-RLW equation(1.1)ut+uxxxxt-uxxt+ux+(up)x=0,x∈[xl,xr],t∈[0,T],
with an initial condition(1.2)u(x,0)=u0(x),x∈[xl,xr],
and boundary conditions(1.3)u(xl,t)=u(xr,t)=0,uxx(xl,t)=uxx(xr,t)=0,t∈[0,T],
where p≥2 is a integer and u0(x) is a known smooth function. When p=2, the equation (1.1) is called usual Rosenau-RLW equation. When p=3, (1.1) is called modified Rosenau-RLW equation.It can be proved easily that the problem (1.1)–(1.3) possesses the following conservative laws:(1.4)Q(t)=∫xlxru(x,t)dx=∫xlxru0(x,t)dx=Q(0),(1.5)E(t)=‖u‖L22+‖ux‖L22+‖uxx‖L22=E(0).As already pointed out by Fei et al. [1], the nonconservative difference schemes may easily show nonlinear blow-up, and the conservative difference schemes perform better than the non-conservative ones. In [2–15], some conservative finite difference schemes were used for Sine-Gordon equation, Cahn-Hilliard equation, Klein-Gordon equation, a system of Schrödinger equation, Zakharov equations, Rosenau equation, GRLW equation, Klein-Gordon-Schrödinger equation, respectively. Numerical results of all the schemes are very good.As far as computational studies are concerned, Zuo et al. [16] have proposed a Crank-Nicolson difference scheme for the Rosenau-RLW equation. The difference scheme in [16] is nonlinear implicit, so it requires heavy iterative calculations and is not suitable for parallel computation. In a recent work [14], we have made some preliminary computation by proposing a conservative linearized difference scheme for GRLW equation which is unconditionally stable and reduces the computational work, and the numerical results are encouraging. In this paper, we continue our work and propose a conservative linearized difference scheme for the general Rosenau-RLW equation which is unconditionally stable and second-order convergent and simulates conservative laws (1.4)-(1.5) at the same time.The remainder of this paper is organized as follows. In Section2, an energy conservative linearized difference scheme for the general Rosenau-RLW equation is described and the discrete conservative laws of the difference scheme are discussed. In Section 3, we show that the scheme is uniquely solvable. In Section 4, convergence and stability of the scheme are proved. In Section 5, numerical experiments are reported.


## 2. An Average Linearized Conservative Scheme and Its Discrete Conservative Law

In this section, we describe a new conservative difference scheme for the problems of (1.1)–(1.3). Let h and τ be the uniform step size in the spatial and temporal direction, respectively. Denote xj=jh(0≤j≤J), tn=nτ(0≤n≤N), ujn≈u(xj,tn) and Zh0={u=(uj)∣u0=uj=0,j=0,1,2,…,J}. Define(2.1)(ujn)x=uj+1n-ujnh,(ujn)x̅=ujn-uj-1nh,(ujn)x̂=uj+1n-uj-1n2h,(ujn)t=ujn+1-ujnτ,(ujn)t̂=ujn+1-ujn-12τ,u̅jn=ujn+1+ujn-12,(un,vn)=h∑j=1J-1ujnvjn,‖un‖2=(un,un),‖un‖∞=max⁡1≤j≤J-1|ujn|,and in the paper,C denotes a general positive constant which may have different values in different occurrences.Notice that(up)x=(p/(p+1))[up-1ux+(up)x]. We consider the following three-level average linearized conservative scheme for the IBV problems (1.1)–(1.3):(2.2)(ujn)t̂+(ujn)xxx̅x̅t̂-(ujn)xx̅t̂+1-θ2(ujn+1+ujn-1)x̂+θ(ujn)x̂+pp+1{(ujn)p-1(u̅jn)x̂+[(ujn)p-1(u̅jn)]x̂}=0,1≤j≤J-1,1≤n≤N-1,(2.3)uj0=u0(xj),1≤j≤J,(2.4)u0n=uJn=0,(u0n)xx̅=(uJn)xx̅=0,0≤n≤N,
where 0≤θ≤1 is a real constant. The scheme (2.2)–(2.4) is three level and linear implicit, so it can be easily implemented. It should be pointed out that we need another suitable two-level scheme (such as C-N scheme) to compute u1. For convenience, the last term of (2.2) is defined by(2.5)Φ(un,u̅n)=pp+1{(ujn)p-1(u̅jn)x̂+[(ujn)p-1(u̅jn)]x̂}.Lemma 2.1 (see [17]).
For any two mesh functions:u,v∈Zh0, one has
(2.6)((uj)x,vj)=-(uj,(vj)x̅),(vj,(uj)xx̅)=-((vj)x,(uj)x),(uj,(uj)xx̅)=-((uj)x,(uj)x)=-‖ux‖2.
Furthermore, if (u0n)xx̅=(uJn)xx̅=0, then
(2.7)(uj,(uj)xxx̅x̅)=‖uxx‖2.Theorem 2.2.
Supposeu0∈H02[xl,xr] and u(x,t)∈C5,3. Then the scheme (2.2)–(2.4) admits the following invariant:
(2.8)Qn=h2∑j=1J-1(ujn+1+ujn)=Qn-1=⋯=Q0,(2.9)En=12(‖un+1‖2+‖un‖2)+12(‖uxxn+1‖2+‖uxxn‖2)+12(‖uxn+1‖2+‖uxn‖2)+θhτ∑j=1J-1(ujn)x̂ujn+1=En-1=⋯=E0.Proof.
Multiplying (2.2) with h, according to the boundary conditions (2.4), then summing up for j from 1 to J-1, we obtain
(2.10)h2∑j=1J-1(ujn+1-ujn-1)=0.
Let
(2.11)Qn=h2∑j=1J-1(ujn+1+ujn).
Then we obtain (2.8) from (2.10).
Taking the inner product of (2.2) with 2u̅n, according to Lemma 2.1, we have
(2.12)12τ(‖un+1‖2-‖un-1‖2)+12τ(‖uxxn+1‖2-‖uxxn-1‖2)+12τ(‖uxn+1‖2-‖uxn-1‖2)+θh∑j=1J-1(ujn)x̂ujn+1-θh∑j=1J-1(ujn-1)x̂ujn+(Φ(un,u̅n),2u̅jn)=0.
Now, computing the last term of the left-hand side in (2.12), we have
(2.13)(Φ(un,u̅n),2u̅jn)=2pp+1h∑j=1J-1{(ujn)p-1(u̅jn)x̂+[(ujn)p-1u̅jn]x̂}u̅jn=pp+1∑j=1J-1[(ujn)p-1(u̅j+1n-u̅j-1n)+(uj+1n)p-1u̅j+1n-(uj-1n)p-1u̅j-1n]u̅jn=pp+1∑j=1J-1[(ujn)p-1u̅j+1nu̅jn-(uj+1n)p-1u̅j+1nu̅jn]-pp+1∑j=1J-1[(uj-1n)p-1u̅jnu̅j-1n-(ujn)p-1u̅jnu̅j-1n]=0.
Substitute (2.13) into (2.12), and we let
(2.14)En=12(‖un+1‖2+‖un‖2)+12(‖uxxn+1‖2+‖uxxn‖2)+12(‖uxn+1‖2+‖uxn‖2)+θhτ∑j=1J-1(ujn)x̂ujn+1.
By the definition of En, (2.9) holds.


## 3. Solvability

In this section, we will prove the solvability of the difference scheme (2.2).Theorem 3.1.
The difference scheme (2.2) is uniquely solvable.Proof.
By the mathematical induction. It is obvious thatu0 is uniquely determined by (2.3). We can choose a second-order method to compute u1 (such as C-N scheme [16]). Assuming that u1,…,un are uniquely solvable, consider un+1 in (2.2) which satisfies
(3.1)12τujn+12τ(ujn)xxx̅x̅-12τ(ujn)xx̅+1-θ2(ujn+1)x̂+p2(p+1){(ujn)p-1(ujn+1)x̂+[(ujn)p-1ujn+1]x̂}=0.
Taking the inner product of (3.1) with un+1, we obtain
(3.2)12τ‖un+1‖2+12τ‖uxxn+1‖2+12τ‖uxn+1‖2+(Ψ(un,un+1),un+1)=0,
where Ψ(un,un+1)=(p/2(p+1))(ujn)p-1(ujn+1)x̂+[(ujn)p-1ujn+1]x̂.
Notice that(3.3)(Ψ(un,un+1),un+1)=ph2(p+1)∑j=1J-1{(ujn)p-1(ujn+1)x̂+[(ujn)p-1ujn+1]x̂}ujn+1.=p4(p+1)∑j=1J-1[(ujn)p-1(uj+1n+1-uj-1n+1)+(uj+1n)p-1uj+1n+1-(uj-1n)p-1uj-1n+1]ujn+1=0.
It follows from (3.2) that
(3.4)12τ‖un+1‖2+12τ‖uxxn+1‖2+12τ‖uxn+1‖2=0.
That is, there uniquely exists trivial solution satisfying (3.1). Hence, ujn+1 in (2.2) is uniquely solvable. This completes the proof of Theorem 3.1.Remark 3.2.
All results above in this paper are correct for IBV problem of the general Rosenau-RLW equation with finite or infinite boundary.


## 4. Convergence and Stability of Finite Difference Scheme

First we will consider the truncation error of the difference scheme of (2.2)–(2.4). Denote vjn=u(xj,tn). We define the truncation error as follows:(4.1)Erjn=(vjn)t̂+(vjn)xxx̅x̅t̂-(vjn)xx̅t̂+1-θ2(vjn+1+vjn-1)x̂+θ(v̅jn)x̂+pp+1{(vjn)p-1(v̅jn)x̂+[(vjn)p-1(v̅jn)]x̂}.
Using Taylor expansion, we obtain that Erjn=O(τ2+h2) holds if τ,h→0.This is that.Lemma 4.1.
Assumeu(x,t) is smooth enough, then the local truncation error of difference scheme (2.2)–(2.4) is O(τ2+h2).Next, we will discuss the convergence and stability of finite difference scheme (2.2)–(2.4). The following two lemmas are introduced.Lemma 4.2 (discrete Sobolev’s inequality [18]).
There exist two constantsC1 and C2 such that
(4.2)‖un‖∞≤C1‖un‖+C2‖uxn‖.Lemma 4.3 (discrete Gronwall inequality [18]).
Supposew(k),ρ(k) are nonnegative mesh functions and ρ(k) is nondecreasing. If C>0 and
(4.3)w(k)≤ρ(k)+Cτ∑l=0k-1w(l),∀k,
then
(4.4)w(k)≤ρ(k)eCτk,∀k.Lemma 4.4.
Supposeu0∈H02[xl,xr], then the solution un of (2.2) satisfies ||un||≤C,||uxn||≤C, which yield ∥un∥∞≤C(n=1,2,…,N).Proof.
It follows from (2.9) that
(4.5)12(‖un+1‖2+‖un‖2)+12(‖uxxn+1‖2+‖uxxn‖2)+12(‖uxn+1‖2+‖uxn‖2)=C-θhτ∑j=1J-1(ujn)x̂ujn+1≤C+12θτ(‖un+1‖2+‖uxn‖2).
Thus
(4.6)12[(1-θτ)‖un+1‖2+‖un‖2]+12(‖uxxn+1‖2+‖uxxn‖2)+12[(1-θτ)‖uxn‖2+‖uxn+1‖2]≤C.
This implies for small τ which satisfies 1-θτ>0, we get
(4.7)‖un‖≤C,‖uxn‖≤C.
Using Lemma 4.2, we obtain
(4.8)‖un‖∞≤C.Remark 4.5.
Lemma4.4 implies that scheme (2.2)–(2.4) is unconditionally stable.Theorem 4.6.
Assume thatu0∈H02[xl,xr] and u(x,t)∈C5,3. Then the solution un of the scheme (2.2)–(2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is O(τ2+h2) by the ∥·∥∞ norm.Proof.
Subtracting (4.1) from (2.2) and letting ejn=vjn-ujn, we have
(4.9)Erjn=(ejn)t̂+(ejn)xxx̅x̅t̂-(ejn)xx̅t̂+(ejn)x̂+1-θ2(ejn+1+ejn-1)x̂+θ(ejn)x̂+pp+1{(vjn)p-1(v̅jn)x̂+[(vjn)p-1v̅jn]x̂}-pp+1{(ujn)p-1(u̅jn)x̂+[(ujn)p-1u̅jn]x̂}.
Taking the inner product in (4.9) with 2e̅n, we obtain
(4.10)(Erjn,2e̅n)=12τ(‖en+1‖2-‖en-1‖2)+12τ(‖exxn+1‖2-‖exxn-1‖2)+12τ(‖exn+1‖2-‖exn-1‖2)+θh∑j=1J-1(ejn)x̂(ejn+1+ejn-1)+(I+II,2e̅n),
where
(4.11)I=pp+1[(vjn)p-1(v̅jn)x̂-(ujn)p-1(u̅jn)x̂],II=pp+1{[(vjn)p-1(v̅jn)]x̂-[(ujn)p-1u̅jn]x̂}.
According to Lemma 4.4, the fifth term of right-hand side of (4.10) is estimated as follows:
(4.12)(I,2e̅n)=2pp+1h∑j=1J-1[(vjn)p-1(v̅jn)x̂-(ujn)p-1(u̅jn)x̂]e̅n=2pp+1h∑j=1J-1(vjn)p-1(e̅jn)x̂e̅n+2pp+1h∑j=1J-1[(vjn)p-1-(ujn)p-1](u̅jn)x̂e̅n=2pp+1h∑j=1J-1(vjn)p-1(e̅jn)x̂e̅n+2pp+1h∑j=1J-1[ejn∑k=0p-2(vjn)p-2-k(ujn)k](u̅jn)x̂e̅n≤C(‖e̅xn‖2+‖en‖2+‖e̅n‖2)≤C(‖exn+1‖2+‖exn-1‖2+‖en+1‖2+‖en‖2+‖en-1‖2),
and similarly we can prove
(4.13)(II,2e̅n)≤C(‖exn+1‖2+‖exn-1‖2+‖en+1‖2+‖en‖2+‖en-1‖2).
In addition, it is obvious that
(4.14)(Erjn,2e̅n)≤‖Ern‖2+12(‖en+1‖2+‖en-1‖2),(4.15)h∑j=1J-1(ejn)x̂(ejn+1+ejn-1)≤‖exn‖2+12(‖en+1‖2+‖en-1‖2).
Substituting (4.12)–(4.15) into (4.10), we get
(4.16)12τ(‖en+1‖2-‖en-1‖2)+12τ(‖exxn+1‖2-‖exxn-1‖2)+12τ(‖exn+1‖2-‖exn-1‖2)≤‖Ern‖2+12(‖en+1‖2+‖en-1‖2)+θ[‖exn‖2+12(‖en+1‖2+‖en-1‖2)]+C(‖en+1‖2+‖en-1‖2+‖en‖2+‖exn+1‖2+‖exn‖2+‖exn-1‖2).
Let Bn=(1/2)(∥en+1∥2+∥en∥2)+(1/2)(∥exxn+1∥2+∥exxn∥2)+(1/2)(∥exn+1∥2+∥exn∥2), then (4.16) can be written as follows:
(4.17)Bn-Bn-1≤τ‖Ern‖2+Cτ(Bn+Bn-1).
Thus
(4.18)(1-Cτ)(Bn-Bn-1)≤2CτBn-1+τ‖Ern‖2.
Hence, for τ sufficiently small, such that 1-Cτ>0, we obtain
(4.19)Bn-Bn-1≤CτBn-1+Cτ‖Ern‖2.
Summing up (4.19) from 1 to n yields
(4.20)Bn≤B0+Cτ∑l=1n‖Erl‖2+Cτ∑l=1nBl.
Choose a second-order method to compute u1 (such as C-N scheme) and notice that
(4.21)τ∑l=1n‖Erl‖2≤nτmax⁡1≤l≤n‖Erl‖2≤T⋅O(τ2+h2)2.
From the discrete initial conditions, we know that e0 is of second-order accuracy, then
(4.22)B0=O(τ2+h2)2.
Then we obtain
(4.23)Bn≤O(τ2+h2)2+Cτ∑l=0n-1Bl.
An application of Lemma 4.3 yields
(4.24)Bn≤O(τ2+h2)2.
Thus
(4.25)‖en‖≤O(τ2+h2),‖exn‖≤O(τ2+h2),‖exxn‖≤O(τ2+h2).
It follows from Lemma 4.2 that
(4.26)‖en‖∞≤O(τ2+h2).
This completes the proof of Theorem 4.6.Similarly, we can prove stability of the difference solution.Theorem 4.7.
Under the conditions of Theorem4.6, the solution of the scheme (3.1)–(2.4) is unconditionally stable by the ∥·∥∞ norm.


## 5. Numerical Experiments

In this section, we conduct some numerical experiments to verify our theoretical results obtained in the previous sections.Consider the general Rosenau-RLW equation(5.1)ut+uxxxxt-uxxt+ux+(up)x=0,x∈[xl,xr],t∈[0,T],
with an initial condition(5.2)u(x,0)=u0(x),x∈[xl,xr],
and boundary conditions(5.3)u(xl,t)=u(xr,t)=0,uxx(xl,t)=uxx(xr,t)=0,t∈[0,T].
The exact solution of the system (5.1)-(5.2) has the following form:(5.4)u(x,t)=eln⁡{(p+3)(3p+1)(p+1)/[2(p2+3)(p2+4p+7)]}/(p-1)sech4/(p-1)[p-14p2+8p+20(x-ct)],
where p≥2 is a integer and c=(p4+4p3+14p2+20p+25)/(p4+4p3+10p2+12p+21).It follows from (5.4) that the initial-boundary value problem (5.1)–(5.3) is consistent to the initial value problem (5.1)-(5.2) for -xl≫0, xr≫0. In the numerical experiments, we take -xl=xr=30,T=10, and consider three cases p=2,4,8, respectively. The errors in the sense of L∞-norm and L2-norm of the numerical solutions are listed on Tables 1, 2, and 3 for three cases p=2,4,8 with θ=1. Tables 1, 2, and 3 verify the second-order convergence and good stability of the numerical solutions.Table 1
The errors of numerical solutions att=10 with p=2 and τ=0.1.
h∥vn-un∥∥vn-un∥∞∥vn4-un4∥/∥vn-un∥∥vn/4-un/4∥∞/∥vn-un∥∞0.251.456039e-41.967455e-4——0.1253.657043e-55.032358e-53.9814653.9096090.06259.084201e-61.257422e-54.0257184.0021240.031252.202821e-63.052964e-64.1238944.118692Table 2
The errors of numerical solutions att=10 with p=4 and τ=0.1.
h∥vn-un∥∥vn-un∥∞∥vn/4-un/4∥/∥vn-un∥∥vn/4-un/4∥∞/∥vn-un∥∞0.252.447510e-43.299748e-4——0.1256.146847e-58.525290e-53.9817323.8705410.06251.526817e-52.119579e-54.0259234.0221610.031253.702240e-65.159101e-64.1240364.108427Table 3
The errors of numerical solutions att=10 with p=8 and τ=0.1.
h∥vn-un∥∥vn-un∥∞∥vn/4-un/4∥/∥vn-un∥∥vn/4-un/4∥∞/∥vn-un∥∞0.252.854206e-43.856020e-4——0.1257.170514e-59.871671e-53.9804763.9061480.06251.781268e-52.465926e-54.0255114.0032310.031254.319352e-65.988124e-64.1239244.118027We have shown in Theorem2.2 that the numerical solution un of the scheme (2.2) satisfies the conservation of discrete mass and energy, respectively. In Tables 4, 5, and 6, the values of (h/2)∑j=1J-1(ujn+1+ujn) and( 1/2)(∥un+1∥2+∥un∥2)+(1/2)(∥uxxn+1∥2+∥uxxn∥2)+(1/2)(∥uxn+1∥2+∥uxn∥2)+θhτ∑j=1J-1(ujn)x̂ujn+1 for the scheme (2.2) are presented for three cases p=2,4,8 under steps h=τ=0.1 with θ=0,0.5 and 1, respectively. It is easy to see from Tables 4, 5, and 6 that the scheme (2.2) preserves the discrete mass and discrete energy very well; thus it can be used to computing for a long time.Table 4
Discrete mass and energy of scheme (2.2) for a few of θ values at different time t with h=τ=0.1 and p=2.
θ=0θ=0.5θ=1tQnEnQnEnQnEn23.79531311.06635043.79531301.06612753.79531291.065904543.79530351.06635043.79530241.06612753.79530241.065904563.79525871.06635043.79525981.06612753.79525911.065904583.79508001.06635043.79508141.06612753.79508291.0659045103.79435811.06635043.79436581.06612753.79437151.0659045Table 5
Discrete mass and energy of scheme (2.2) for a few of θ values at different time t with h=τ=0.1 and p=4.
θ=0θ=0.5θ=1tQnEnQnEnQnEn26.26556062.86767426.26556032.86708796.26556002.866501646.26534402.86767426.26534372.86708796.26534352.866501666.26480792.86767426.26480832.86708796.26480862.866501686.26355452.86767426.26355702.86708796.26355952.8665016106.26064412.86767426.26065292.86708796.26066162.8665016Table 6
Discrete mass and energy of scheme (2.2) for a few of θ values at different time t with h=τ=0.1 and p=8.
θ=0θ=0.5θ=1tQnEnQnEnQnEn29.72022654.73512699.72021054.73459609.72019544.734065049.71400744.73512709.71397774.73459609.71394864.734065069.70243464.73512709.70239764.73459609.70236104.734065089.68349254.73512709.68345664.73459609.68342094.7340650109.65360974.73512709.65358504.73459609.65356024.7340650We make a comparison between C-N scheme [16] and our scheme with θ=0,0.5,1 under the meshes h=τ=0.1 in Figures 1 and 2 when p=4. It is obvious from Figures 1 and 2 that our scheme performs better than C-N scheme [16] in the numerical precision when θ=0.5 and 1. Figures 1 and 2 also show that numerical precision of the scheme (2.2) depends on the choice of parameter θ. The curves of the solitary waves with time computed by the scheme (2.2) with θ=0 for p=2 and θ=1 for p=4 under mesh sizes of h=τ=0.1 are given in Figures 3 and 4, respectively; the waves at t=5,10 agree with the ones at t=0 quite well, which also demonstrate the accuracy of the scheme in present paper.Figure 1
Errors in the sense of∥en∥∞ computed by the scheme (2.2) when h=τ=0.1 and p=4.Figure 2
Errors in the sense of∥en∥2 computed by the scheme (2.2) when h=τ=0.1 and p=4.Figure 3
Exact solutions ofu(x,t) at t=0 and numerical solutions computed by the scheme (2.2) at t=5,10 with θ=0 for p=2.Figure 4
Exact solutions ofu(x,t) at t=0 and numerical solutions computed by the scheme (2.2) at t=5,10 with θ=1 for p=4.From the numerical results, the scheme of this paper is accurate and efficient.



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