# Explicit Multistep Mixed Finite Element Method for RLW Equation

Authors: Yang Liu; Hong Li; Yanwei Du; Jinfeng Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

An explicit multistep mixed finite element method is proposed and discussed for regularized long wave (RLW) equation. The spatial direction is approximated by the mixed Galerkin method using mixed linear space finite elements, and the time direction is discretized by the explicit multistep method. The optimal error estimates inL2 and H1 norms for the scalar unknown u and its flux q=ux
based on time explicit multistep method are derived. Some numerical results are given to verify our theoretical analysis and illustrate the efficiency of our method.


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## Body

## 1. Introduction

In this paper, we consider the following initial boundary problem of RLW equation:(1)ut+γux+δuux-βuxxt=0,(x,t)∈I×J,u(a,t)=u(b,t)=0,t∈J-,u(x,0)=u0(x),x∈I-,
where I=(a,b) is a bounded open interval, J=(0,T] with 0<T<∞. The initial value u0(x) is given function, and the coefficients β, γ, δ are all positive constants.In recent years, large nonlinear phenomena are found in many research fields, for example, physics, biology, fluid dynamics, and so forth. These phenomena can be described by the mathematical model of some nonlinear evolution equations. In particular, some attention has also been paid to nonlinear RLW equations [1, 2] which play a very important role in the study of nonlinear dispersive waves. Solitary waves are wave packet or pulses, which propagate in nonlinear dispersive media. Due to dynamical balance between the nonlinear and dispersive effects, these waves retain an unchanged waveform. A soliton is a very special type of solitary wave, which also keeps its waveform after collision with other solitons. The regularized long wave (RLW) equation is an alternative description of nonlinear dispersive waves to the more usual Korteweg-de Vries (KdV) equation. Mathematical theories and numerical methods for (1) were considered in [1–24]. The existence and uniqueness of the solution of RLW equation are discussed in [5]. Their analytical solutions were found under restricted initial and boundary conditions, and therefore they got interest from a numerical point of view. Several numerical methods for the solution of the RLW equation have been introduced in the literature. These include a variety of difference methods [5–8, 17], finite element methods based on Galerkin and collocation principles [9–13], mixed finite element methods [14–16], meshfree method [18], adomian decomposition method [19], and so on.In [25], Chatzipantelidis studied the explicit multistep methods for some nonlinear partial differential equations and discussed some mathematical theories. Akrivis et al. [26] studied the multistep method for some nonlinear evolution equations. Mei and Chen [20] presented the explicit multistep method based on Galerkin method for regularized long wave (RLW) equation. In this paper, our purpose is to propose and study an explicit multistep mixed method, which combines a mixed Galerkin method in the spatial direction and the explicit multistep method in the time direction, for RLW equation. We derive optimal error estimates in L2 and H1 norms for the scalar unknown u and its flux q=ux for the fully discrete explicit multistep mixed scheme and compare our method’s accuracy with some other numerical schemes. Compared to the numerical methods in [20, 25, 26], we not only obtain the approximation solution for u, but also get the approximation solution for q=ux.The layout of the paper is as follows. In Section2, an explicit multistep mixed scheme and numerical process are given. The optimal error estimates in L2 and H1 norms for the scalar unknown u and its flux q=ux for the fully discrete explicit multistep mixed scheme are proved in Section 3. In Section 4, some numerical results are shown to confirm our theoretical analysis. Finally, some concluding remarks are given in Section 5. Throughout this paper, C will denote a generic positive constant which does not depend on the spatial mesh parameter h or time discretization parameter Δt.


## 2. The Mixed Numerical Scheme

With the auxiliary variableq=ux, we reformulate (1) as the following first-order coupled system:
(2)ux=q,ut+γq+δuq-βqxt=0.
We consider the following mixed weak formulation of (2). Find {u,q}:[0,T]↦H01×H1 satisfying:
(3)(ux,vx)=(q,vx),∀v∈H01,(4)(qt,w)+β(qxt,wx)=γ(q,wx)+δ(uq,wx),∀w∈H1.
Noting the Dirichlet boundary conditions ut(a,t)=ut(b,t)=0 and qt=uxt, we can get (ut,-wx)=(uxt,w)=(qt,w) easily and then get the scheme (4).LetVh and Wh be finite dimensional subspaces of H01 and H1, respectively, defined by
(5)Vh={vh|vh∈C0(I-),vh|Ij∈Pk(Ij),c∀Ij∈Th,vh(a)=vh(b)=0vh∈C0(I-),vh|Ij}⊂H01,Wh={wh|wh∈C0(I-),wh|Ij∈Pr(Ij),∀Ij∈Th}⊂H1,
where Th is a partition of I-=[a,b] into N subintervals Ij=[xj,xj+1], j=0,1,2,…,(N-1), hj=hj+1-hj, h=max0≤j≤N-1hj, and Pm(Ij) denotes the polynomials of degree less than or equal to m in Ij.The semidiscrete mixed finite element method for (3) and (4) consists in determining {uh,qh}:[0,T]↦Vh×Wh such that
(6)(uxh,vxh)=(qh,vxh),∀vh∈Vh,(7)(qth,wh)+β(qxth,wxh)=γ(qh,wxh)+δ(uhqh,wxh),∀wh∈Wh.In the following discussion, we will give an explicit multistep mixed scheme. We take linear finite element spacesVh=span{φ0,φ1,…,φN} and Wh=span{φ0,φ1,…,φN}, and then uh and qh can be expressed as the following formulation:
(8)uh(x,t)=∑i=0N‍ui(t)φi(x),(x,t)∈Ω×J,qh(x,t)=∑i=0N‍qi(t)φi(x),(x,t)∈Ω×J.
Substitute (8) into (6) and (7), and take vh=φj and wh=φj in (6) and (7), respectively, to obtain
(9)∑i=0N‍[(∫ab‍φi·φjdx+β∫ab‍φix·φjxdx)∂qi∂t∑i=0N‍c-(γ∫ab‍φi·φjxdx+δ∫ab‍(∑i=0N‍uiφi)φi·φjxdx)qi]=0,∑i=0N‍[(∫ab‍φix·φjxdx)ui-(∫ab‍φi·φjxdx)qi]=0,
where j=0,1,2,…,N.We subdivide the space variable domain[a,b] into uniform subintervals with N+1 grid points xk,k=0,…,N, such that a=x0<x1<⋯<xN=b, h=xk+1-xk=(b-a)/N. Using the local coordinate transformation x=xk+μh, 0≤μ≤1, we transform a subinterval [xk,xk+1] into a standard interval [0,1]. Furthemore, we have
(10)∑i=kk+1‍[(∑i=0N‍uiφi)(∫01‍φi·φjdx+βh2∫01‍φix·φjxdx)∂qi∂t∑i=kk+1‍c-1h(γ∫01‍φi·φjxdx+δ∫01‍(∑i=0N‍uiφi)φi·φjxdx)qi]=0,∑i=kk+1‍[1h2(∫01‍φix·φjxdx)ui-1h(∫ab‍φi·φjxdx)qi]=0.
We take linear basis functions defined as follows:
(11)L1=1-μ,L2=μ,
and then the variables u and q over the element [xk,xk+1] are written as
(12)ue=∑j=12‍Ljuj,qe=∑j=12‍Ljqj.
Then, we get the following equations:
(13)∑i=12‍[(∑i=12‍uiLi)(∫01‍Li·Ljdμ+βh2∫01‍Liμ·Ljμdμ)∂qi∂t∑i=12‍c-1h(γ∫01‍Li·Ljμdμ+δ∫01‍(∑i=12‍uiLi)Li·Ljμdμ)qi]=0,∑i=12‍[1h2(∫01‍Liμ·Ljμdμ)ui-1h(∫ab‍Li·Ljμdμ)qi]=0.
Then, the system (13) has the following matrix form:
(14)(Aije+βBije)∂qe∂t-(γCije+δDije(ue))qe=0,Bijeue-Cijeqe=0,
with the following element matrices:
(15)ue=(u1,u2)T,qe=(q1,q2)T,Aije=∫01‍Li·Ljdμ,Bije=1h2∫01‍Liμ·Ljμdμ,Cije=1h∫01‍Li·Ljμdμ,Djke=δh∫01‍(∑i=12‍uiLi)Liμ·Ljdμ.
Assembling contributions from all elements, we obtain the following coupled system of nonlinear matrix equations:
(16)(A+βB)∂qh∂t-(γC+δD(uh))qh=0,Buh-Cqh=0.
To formulate a fully discrete scheme, we consider a uniform partition of J-=[0,T] with time step length Δt=T/N, N∈Z+, and time levels tn=nΔt, n=0,…,N. We now discuss a fully discrete scheme based on a linear explicit multistep method. We now define Un∈Vh and Zn∈Wh as approximations to u(tn) and q(tn), respectively, and formulate the following fully discrete linear explicit multistep mixed scheme:
(17)(A+βB)∑i=0p‍αiZn+i=Δt∑i=0p-1‍σi[(γC+δD(Un+i))Zn+i],BUn+p=CZn+p,
with given the initial approximations U0,…,Up-1 and Z0,…,Zp-1. In the explicit multistep mixed system (17), the parameter variable αi and σi is described by the coefficients of the term χi, for the following polynomials α(χ) and σ(χ), respectively:
(18)α(χ)≔∑j=1p‍1jχp-j(χ-1)j,σ(χ)≔χp-(χ-1)p.In this paper, we consider the explicit 2-step mixed method for the RLW equation. Forp=2, we obtained easily
(19)α0=32,α1=-2,α2=12,σ0=-1,σ1=2.
Substituting (19) into (17), we obtain the following 2-step mixed scheme:
(20)(A+βB)(32Zn+2-2Zn+1+12Zn)=Δt[γC(2Zn+1-Zn)+δ(Zn+12D(Un+1)Zn+1-D(Un)ZnZn+1)],BUn+2=CZn+2.Remark 1.
There have been many numerical schemes for the RLW equation, but we have not seen the related research on explicit multistep mixed element method for RLW equation in the literature. From the viewpoint of numerical theory, we propose a mixed element scheme (6) and (7), which is different from some other mixed finite element methods in [14–16], for the RLW equation and derive some a priori error estimates based on the explicit multistep mixed element method. From the perspective of numerical calculation, our method is efficient for RLW equation.


## 3. Two-Step Mixed Scheme and Optimal Error Estimates

### 3.1. Two-Step Mixed Scheme and Some Lemmas

In this section, we will discuss some a priori error estimates based on explicit 2-step mixed finite element method for the RLW equation. For the fully discrete procedure, let0=t0<t1<⋯<tN=T  be a given partition of the time interval [0,T] with step length Δt=T/N, for some positive integer N. For a smooth function ϕ on [0,T], define ϕn=ϕ(tn).The system (3) and (4) has the following formulation at t=tn+1:
(21)(uxn+1,vx)=(qn+1,vx),∀v∈H01,(qtn+1,w)+β(qxtn+1,wx)=γ(qn+1,wx)+δ(un+1qn+1,wx),∀w∈H1.
Based on system (17), we get an equivalent formulation for system (21) as
(22)(uxn+1,vx)=(qn+1,vx),∀v∈H01,(23)(3qn+1-4qn+qn-12Δt,w)+β(3qxn+1-4qxn+qxn-12Δt,wx)=-(τn+1,w)-β(κ1n+1,wx)+γ(2qn-qn-1,wx)+δ(2unqn-un-1qn-1,wx)+γ(R1n+1,wx)+δ(R2n+1,wx),∀w∈H1,
where
(24)τn+1=qtn+1-3qn+1-4qn+qn-12Δt,κ1n+1=qxt(tn+1)-3qxn+1-4qxn+qxn-12Δt,R1n+1=qn+1-(2qn-qn-1),R2n+1=un+1qn+1-(2unqn-un-1qn-1).
We now find a pair {Un+1,Zn+1} in Vh×Wh satisfying
(25)(Uxn+1,vxh)=(Zn+1,vxh),∀vh∈Vh,(26)(3Zn+1-4Zn+Zn-12Δt,wh)+β(3Zxn+1-4Zxn+Zxn-12Δt,wxh)=γ(2Zn-Zn-1,wxh)+δ(2UnZn-Un-1Zn-1,wxh),γ(2Zn-Zn-1,wxh)+∀wh∈Wh.For the theoretical analysis of a priori error estimates, we define the following projections.Lemma 2 (see [15, 27, 28]).
One defines the elliptic projectionu~h∈Vh by
(27)(ux-u~xh,vxh)=0,vh∈Vh.
With η=u-u~h, the following estimates are well known for j=0,1:
(28)∥η∥j≤Chk+1-j∥u∥k+1.Lemma 3 (see [15, 27, 28]).
Furthermore, one also defines a Ritz projectionq~h∈Wh of q as the solution of
(29)A(q-q~h,wh)=0,wh∈Wh,
where A(q,w)=(qx,wx)+λ(q,w), and λ is taken appropriately so that
(30)A(w,w)≥μ0∥w∥12,w∈H1,
where μ0 is a positive constant. Moreover, it is easy to verify that A(·,·) is bounded.
Withρ=q-q~h, the following estimates hold:
(31)∥∂iρ∂ti∥j≤Chr+1-j∥∂iq∂ti∥r+1,i=0,1,2,3;j=0,1.For fully discrete error estimates, we now write the errors as(32)u(tn)-Un=(u(tn)-u~h(tn))+(u~h(tn)-Un)=ηn+ςn,q(tn)-Zn=(q(tn)-q~h(tn))+(q~h(tn)-Zn)=ρn+ξn.
Combine (27), (29), (22), (23), (25), and (26) at t=tn+1 to get the following error equations:
(33)(ςxn+1,vxh)=(ρn+1+ξn+1,vxh),∀vh∈Vh,(34)(3ξn+1-4ξn+ξn-12Δt,wh)+β(3ξxn+1-4ξxn+ξxn-12Δt,wxh)=-((1-βλ)3ρn+1-4ρn+ρn-12Δt+τn+1,wh)-β(κ2n+1,wxh)+γ(2ξn-ξn-1,wxh)+δ(wxh2(u(tn)q(tn)-UnZn)cc-(u(tn-1)q(tn-1)-Un-1Zn-1),wxh)+γ(R1n+1,wxh)+δ(R2n+1,wxh),∀wh∈Wh,
where
(35)κ2n+1=q~xth(tn+1)-3q~xh,n+1-4q~xh,n+q~xh,n-12Δt.Lemma 4.
Forτn+1, κ2n+1, R1n+1, and R2n+1, the following estimates hold:
(36)∥τn+1∥≤CΔt2∥qttt∥L∞(L2),∥κ2n+1∥≤CΔt2(hr∥qxttt∥L∞(Hr+1)+∥qxttt∥L∞(L2)),∥R1n+1∥≤CΔt2∥qtt∥L∞(L2),∥R2n+1∥≤CΔt2(∥uqtt∥L∞(L2)+∥utqt∥L∞(L2)+∥uttq∥L∞(L2)).Proof.
Using the Taylor expansion, we have(37)q(tn)=q(tn+1)+qt(tn+1)(tn-tn+1)+qtt(tΔ1)2(tn-tn+1)2,tn<tΔ1<tn+1,(38)q(tn-1)=q(tn+1)+qt(tn+1)(tn-1-tn+1)+qtt(tΔ2)2(tn-1-tn+1)2,tn-1<tΔ2<tn+1.
Combining (37) and (38) and noting that -2Δt=2(tn-tn+1)=tn-1-tn+1, we obtain
(39)q(tn+1)=2q(tn)-q(tn-1)+(qtt(tΔ1)-2qtt(tΔ2))Δt2.
From (39), we have
(40)∥q(tn+1)-(2q(tn)-q(tn-1))∥≤CΔt2∥qtt∥L∞(L2).
Using a similar estimate as the one for ∥R1n+1∥, we have
(41)u(tn+1)q(tn+1)=2u(tn)q(tn)-u(tn-1)q(tn-1)+((uq)tt(tΔ3)-2(uq)tt(tΔ4))Δt2,
where tn<tΔ3<tn+1, tn-1<tΔ4<tn+1.
From (41), we have
(42)∥u(tn+1)q(tn+1)-(2u(tn)q(tn)-u(tn-1)q(tn-1))∥≤CΔt2(∥uqtt∥L∞(L2)+∥utqt∥L∞(L2)+∥uttq∥L∞(L2)).
Using the Taylor expansion and noting that -2Δt=2(tn-tn+1)=tn-1-tn+1, we have
(43)4q(tn)=4q(tn+1)-4qt(tn+1)Δt+2qtt(tn+1)Δt2-2qttt(tΔ5)3Δt3,tn<tΔ5<tn+1,q(tn-1)=q(tn+1)-2qt(tn+1)Δt+2qtt(tn+1)2Δt2-4qttt(tΔ6)3Δt3,tn-1<tΔ6<tn+1.
Using (43), we obtain
(44)qtn+1=3qn+1-4qn+qn-12Δt-(2qttt(tΔ5)3-4qttt(tΔ6)3)Δt2.
By (44), we have
(45)∥τn+1∥≤CΔt2∥qttt(t)∥L∞(L2).
Using the similar method to the estimate for ∥τn+1∥ and (31), we obtain
(46)∥κ2n+1∥≤CΔt2∥q~xttth∥L∞(L2)≤CΔt2(∥ρxttt∥L∞(L2)+∥qxttt∥L∞(L2))≤CΔt2(hr∥qxttt∥L∞(Hr+1)+∥qxttt∥L∞(L2)).


### 3.2. Optimal Error Estimates

In this subsection, we derive the fully discrete optimal error estimates and obtain the following theorem.Theorem 5.
Assuming thatU0, U1∈Vh and Z0, Z1∈Wh are given, then for 1≤J≤M, j=0,1, one has
(47)∥uJ+1-UJ+1∥j≤C(hmin(k+1-j,r+1)+Δt2),∥qJ+1-ZJ+1∥j+∥2(qJ+1-ZJ+1)-(qJ-ZJ)∥j≤C(hmin(k+1,r+1-j)+Δt2).Proof.
Takingvh=ςn+1 in (33) and using Cauchy-Schwarz’s inequality and Poincaré’s inequality, we get
(48)∥ςn+1∥≤C∥ςxn+1∥≤C(∥ρn+1∥+∥ξn+1∥).
Set wh=ξn+1 in (34) to obtain
(49)(3ξn+1-4ξn+ξn-12Δt,ξn+1)+β(3ξxn+1-4ξxn+ξxn-12Δt,ξxn+1)=-((1-βλ)3ρn+1-4ρn+ρn-12Δt+τn+1,ξn+1)-β(κn+1,ξxn+1)+γ(2ρn-ρn-1,ξxn+1)+γ(2ξn-ξn-1,ξxn+1)+γ(R1n+1,ξxn+1)+δ(R2n+1,ξxn+1)+δ(ξxn+12(u(tn)q(tn)-UnZn)cc-(u(tn-1)q(tn-1)-Un-1Zn-1),ξxn+1).
Use (48) as well as the Cauchy-Schwarz and Young’s inequalities to obtain
(50)(3ξn+1-4ξn+ξn-12Δt,ξn+1)+β(3ξxn+1-4ξxn+ξxn-12Δt,ξxn+1)≤C(∥ξn+1∥2+∥ξxn+1∥2+∥ξn∥2+∥2ρn-ρn-1∥2ccc+∥2ξn-ξn-1∥2+∥3ρn+1-4ρn+ρn-12Δt∥2ccc+∥τn+1∥2+∥κn+1∥2+∥R1n+1∥2+∥R2n+1∥2∥2ρn-ρn-1∥2)+δ|(Zn-12(u(tn)q(tn)-UnZn)ccc-(u(tn-1)q(tn-1)-Un-1Zn-1),ξxn+1)|.
Note that
(51)(a)(3ξn+1-4ξn+ξn-12Δt,ξn+1)(a)=14Δt[(∥ξn+1∥2+∥2ξn+1-ξn∥2)(a)14Δtc-(∥ξn∥2+∥2ξn-ξn-1∥2)+∥ξn+1-2ξn+ξn-1∥2],(b)β(3ξxn+1-4ξxn+ξxn-12Δt,ξxn+1)(b)=14Δt[(∥ξxn+1∥2+∥2ξxn+1-ξxn∥2)14Δtc(b)c-(∥ξxn∥2+∥2ξxn-ξxn-1∥2)14Δtc(b)c+∥ξxn+1-2ξxn+ξxn-1∥2],(c)∥3ρn+1-4ρn+ρn-12Δt∥2≤32Δt∫tntn+1‍∥ρt∥2ds(c)∥3ρn+1-4ρn+ρn-12Δt∥2≤+12Δt∫tn-1tn‍∥ρt∥2ds,(52)|(ξxn+12(u(tn)q(tn)-UnZn)cc-(u(tn-1)q(tn-1)-Un-1Zn-1),ξxn+1)|≤|(2u(tn)(q(tn)-Zn)-u(tn)(q(tn-1)-Zn-1),ξxn+1)|+|((u(tn)-u(tn-1))(q(tn-1)-Zn-1),ξxn+1)|+|(2(u(tn)-Un)Zn-(u(tn-1)-Un-1)Zn,ξxn+1)|+|((u(tn-1)-Un-1)(Zn-Zn-1),ξxn+1)|=˙T1+T2+T3+T4.
We now estimate T1, T2, T3, and T4 as
(53)T1≤|(u(tn)(2ξn-ξn-1),ξxn+1)|+|(u(tn)(2ρn-ρn-1),ξxn+1)|≤∥u(tn)∥∞∥2ξn-ξn-1∥∥ξxn+1∥+∥u(tn)∥∞∥2ρn-ρn-1∥∥ξxn+1∥,T2=|(∫tn-1tn‍utds(ρn-1+ξn-1),ξxn+1)|≤Δt∥ut∥∞∥ρn-1+ξn-1∥∥ξxn+1∥,T3≤|(Zn(2ςn-ςn-1),ξxn+1)|+|(Zn(2ηn-ηn-1),ξxn+1)|≤∥Zn∥∞∥2ςn-ςn-1∥∥ξxn+1∥+∥Zn∥∞∥2ηn-ηn-1∥∥ξxn+1∥,T4≤|(∫tn-1tn‍Ztds(ηn-1+ςn-1),ξxn+1)|≤Δt∥Zt∥∞∥ηn-1+ςn-1∥∥ξxn+1∥.
Substituting (53) into (52), we obtain
(54)|(ξxn+12(u(tn)q(tn)-UnZn)cc-(u(tn-1)q(tn-1)-Un-1Zn-1),ξxn+1)|≤C(∥2ξn-ξn-1∥2+∥ξn-1∥2ccc+∥ρn∥2+∥ρn-1∥2+∥ςn∥2+∥ςn-1∥2ccc+∥ηn∥2+∥ηn-1∥2+∥ξxn+1∥2).
Substituting (36), (51), and (54) into (50), using (48), and summing from n=1,2,…,J, the resulting inequality becomes
(55)(1-CΔt)(∥ξJ+1∥2+∥2ξJ+1-ξJ∥2+∥ξxJ+1∥2+∥2ξxJ+1-ξxJ∥2)≤CΔt∑n=1J‍(∥ηn-1∥2+∥ρn-1∥2)+C∫0tJ+1‍∥ρt∥2ds+CΔt4(∥qtt∥L∞(L2)2+∥qttt∥L∞(L2)2c+∥uqtt∥L∞(L2)+∥utqt∥L∞(L2)+∥uttq∥L∞(L2)c+hr∥qxttt∥L∞(Hr+1)+∥qxttt∥L∞(L2)∥qtt∥L∞(L2)2)+CΔt∑n=1J‍(∥ξn∥2+∥ξxn∥2+∥2ξn-ξn-1∥2+∥2ξxn-ξxn-1∥2).
Choose Δt0 in such a way that for 0<Δt≤Δt0, (1-CΔt)>0. Then, as an application of Gronwall’s lemma, we obtain
(56)∥ξJ+1∥2+∥2ξJ+1-ξJ∥2+∥ξxJ+1∥2+∥2ξxJ+1-ξxJ∥2≤CΔt∑n=1J‍(∥ηn-1∥2+∥ρn-1∥2)+C∫0tJ+1‍∥ρt∥2ds+CΔt4(∥qtt∥L∞(L2)2+∥qttt∥L∞(L2)2+∥uqtt∥L∞(L2)+CΔt4cc+∥utqt∥L∞(L2)+∥uttq∥L∞(L2)+CΔt4cc+hr∥qxttt∥L∞(Hr+1)+∥qxttt∥L∞(L2)∥qtt∥L∞(L2)2).
Combine (28), (31), and (56) with the triangle inequality to complete the L2 and H1 error estimates for q. Furthermore, use (48) and the triangle inequality to complete the optimal error estimates for ∥u(tn)-Un∥ and ∥u(tn)-Un∥1.Remark 6.
Compared to a variety of difference methods in [17], our method is studied based on mixed element scheme (6) and (7).Remark 7.
Although some convergence proofs of multistep methods for RLW/BBM are provided in [20, 25, 29], our convergence results of multistep methods are proved based on a mixed finite element scheme. Based on the current discussion, we have to provide the detailed proofs for multistep mixed finite element methods in this paper.


## 3.1. Two-Step Mixed Scheme and Some Lemmas

In this section, we will discuss some a priori error estimates based on explicit 2-step mixed finite element method for the RLW equation. For the fully discrete procedure, let0=t0<t1<⋯<tN=T  be a given partition of the time interval [0,T] with step length Δt=T/N, for some positive integer N. For a smooth function ϕ on [0,T], define ϕn=ϕ(tn).The system (3) and (4) has the following formulation at t=tn+1:
(21)(uxn+1,vx)=(qn+1,vx),∀v∈H01,(qtn+1,w)+β(qxtn+1,wx)=γ(qn+1,wx)+δ(un+1qn+1,wx),∀w∈H1.
Based on system (17), we get an equivalent formulation for system (21) as
(22)(uxn+1,vx)=(qn+1,vx),∀v∈H01,(23)(3qn+1-4qn+qn-12Δt,w)+β(3qxn+1-4qxn+qxn-12Δt,wx)=-(τn+1,w)-β(κ1n+1,wx)+γ(2qn-qn-1,wx)+δ(2unqn-un-1qn-1,wx)+γ(R1n+1,wx)+δ(R2n+1,wx),∀w∈H1,
where
(24)τn+1=qtn+1-3qn+1-4qn+qn-12Δt,κ1n+1=qxt(tn+1)-3qxn+1-4qxn+qxn-12Δt,R1n+1=qn+1-(2qn-qn-1),R2n+1=un+1qn+1-(2unqn-un-1qn-1).
We now find a pair {Un+1,Zn+1} in Vh×Wh satisfying
(25)(Uxn+1,vxh)=(Zn+1,vxh),∀vh∈Vh,(26)(3Zn+1-4Zn+Zn-12Δt,wh)+β(3Zxn+1-4Zxn+Zxn-12Δt,wxh)=γ(2Zn-Zn-1,wxh)+δ(2UnZn-Un-1Zn-1,wxh),γ(2Zn-Zn-1,wxh)+∀wh∈Wh.For the theoretical analysis of a priori error estimates, we define the following projections.Lemma 2 (see [15, 27, 28]).
One defines the elliptic projectionu~h∈Vh by
(27)(ux-u~xh,vxh)=0,vh∈Vh.
With η=u-u~h, the following estimates are well known for j=0,1:
(28)∥η∥j≤Chk+1-j∥u∥k+1.Lemma 3 (see [15, 27, 28]).
Furthermore, one also defines a Ritz projectionq~h∈Wh of q as the solution of
(29)A(q-q~h,wh)=0,wh∈Wh,
where A(q,w)=(qx,wx)+λ(q,w), and λ is taken appropriately so that
(30)A(w,w)≥μ0∥w∥12,w∈H1,
where μ0 is a positive constant. Moreover, it is easy to verify that A(·,·) is bounded.
Withρ=q-q~h, the following estimates hold:
(31)∥∂iρ∂ti∥j≤Chr+1-j∥∂iq∂ti∥r+1,i=0,1,2,3;j=0,1.For fully discrete error estimates, we now write the errors as(32)u(tn)-Un=(u(tn)-u~h(tn))+(u~h(tn)-Un)=ηn+ςn,q(tn)-Zn=(q(tn)-q~h(tn))+(q~h(tn)-Zn)=ρn+ξn.
Combine (27), (29), (22), (23), (25), and (26) at t=tn+1 to get the following error equations:
(33)(ςxn+1,vxh)=(ρn+1+ξn+1,vxh),∀vh∈Vh,(34)(3ξn+1-4ξn+ξn-12Δt,wh)+β(3ξxn+1-4ξxn+ξxn-12Δt,wxh)=-((1-βλ)3ρn+1-4ρn+ρn-12Δt+τn+1,wh)-β(κ2n+1,wxh)+γ(2ξn-ξn-1,wxh)+δ(wxh2(u(tn)q(tn)-UnZn)cc-(u(tn-1)q(tn-1)-Un-1Zn-1),wxh)+γ(R1n+1,wxh)+δ(R2n+1,wxh),∀wh∈Wh,
where
(35)κ2n+1=q~xth(tn+1)-3q~xh,n+1-4q~xh,n+q~xh,n-12Δt.Lemma 4.
Forτn+1, κ2n+1, R1n+1, and R2n+1, the following estimates hold:
(36)∥τn+1∥≤CΔt2∥qttt∥L∞(L2),∥κ2n+1∥≤CΔt2(hr∥qxttt∥L∞(Hr+1)+∥qxttt∥L∞(L2)),∥R1n+1∥≤CΔt2∥qtt∥L∞(L2),∥R2n+1∥≤CΔt2(∥uqtt∥L∞(L2)+∥utqt∥L∞(L2)+∥uttq∥L∞(L2)).Proof.
Using the Taylor expansion, we have(37)q(tn)=q(tn+1)+qt(tn+1)(tn-tn+1)+qtt(tΔ1)2(tn-tn+1)2,tn<tΔ1<tn+1,(38)q(tn-1)=q(tn+1)+qt(tn+1)(tn-1-tn+1)+qtt(tΔ2)2(tn-1-tn+1)2,tn-1<tΔ2<tn+1.
Combining (37) and (38) and noting that -2Δt=2(tn-tn+1)=tn-1-tn+1, we obtain
(39)q(tn+1)=2q(tn)-q(tn-1)+(qtt(tΔ1)-2qtt(tΔ2))Δt2.
From (39), we have
(40)∥q(tn+1)-(2q(tn)-q(tn-1))∥≤CΔt2∥qtt∥L∞(L2).
Using a similar estimate as the one for ∥R1n+1∥, we have
(41)u(tn+1)q(tn+1)=2u(tn)q(tn)-u(tn-1)q(tn-1)+((uq)tt(tΔ3)-2(uq)tt(tΔ4))Δt2,
where tn<tΔ3<tn+1, tn-1<tΔ4<tn+1.
From (41), we have
(42)∥u(tn+1)q(tn+1)-(2u(tn)q(tn)-u(tn-1)q(tn-1))∥≤CΔt2(∥uqtt∥L∞(L2)+∥utqt∥L∞(L2)+∥uttq∥L∞(L2)).
Using the Taylor expansion and noting that -2Δt=2(tn-tn+1)=tn-1-tn+1, we have
(43)4q(tn)=4q(tn+1)-4qt(tn+1)Δt+2qtt(tn+1)Δt2-2qttt(tΔ5)3Δt3,tn<tΔ5<tn+1,q(tn-1)=q(tn+1)-2qt(tn+1)Δt+2qtt(tn+1)2Δt2-4qttt(tΔ6)3Δt3,tn-1<tΔ6<tn+1.
Using (43), we obtain
(44)qtn+1=3qn+1-4qn+qn-12Δt-(2qttt(tΔ5)3-4qttt(tΔ6)3)Δt2.
By (44), we have
(45)∥τn+1∥≤CΔt2∥qttt(t)∥L∞(L2).
Using the similar method to the estimate for ∥τn+1∥ and (31), we obtain
(46)∥κ2n+1∥≤CΔt2∥q~xttth∥L∞(L2)≤CΔt2(∥ρxttt∥L∞(L2)+∥qxttt∥L∞(L2))≤CΔt2(hr∥qxttt∥L∞(Hr+1)+∥qxttt∥L∞(L2)).


## 3.2. Optimal Error Estimates

In this subsection, we derive the fully discrete optimal error estimates and obtain the following theorem.Theorem 5.
Assuming thatU0, U1∈Vh and Z0, Z1∈Wh are given, then for 1≤J≤M, j=0,1, one has
(47)∥uJ+1-UJ+1∥j≤C(hmin(k+1-j,r+1)+Δt2),∥qJ+1-ZJ+1∥j+∥2(qJ+1-ZJ+1)-(qJ-ZJ)∥j≤C(hmin(k+1,r+1-j)+Δt2).Proof.
Takingvh=ςn+1 in (33) and using Cauchy-Schwarz’s inequality and Poincaré’s inequality, we get
(48)∥ςn+1∥≤C∥ςxn+1∥≤C(∥ρn+1∥+∥ξn+1∥).
Set wh=ξn+1 in (34) to obtain
(49)(3ξn+1-4ξn+ξn-12Δt,ξn+1)+β(3ξxn+1-4ξxn+ξxn-12Δt,ξxn+1)=-((1-βλ)3ρn+1-4ρn+ρn-12Δt+τn+1,ξn+1)-β(κn+1,ξxn+1)+γ(2ρn-ρn-1,ξxn+1)+γ(2ξn-ξn-1,ξxn+1)+γ(R1n+1,ξxn+1)+δ(R2n+1,ξxn+1)+δ(ξxn+12(u(tn)q(tn)-UnZn)cc-(u(tn-1)q(tn-1)-Un-1Zn-1),ξxn+1).
Use (48) as well as the Cauchy-Schwarz and Young’s inequalities to obtain
(50)(3ξn+1-4ξn+ξn-12Δt,ξn+1)+β(3ξxn+1-4ξxn+ξxn-12Δt,ξxn+1)≤C(∥ξn+1∥2+∥ξxn+1∥2+∥ξn∥2+∥2ρn-ρn-1∥2ccc+∥2ξn-ξn-1∥2+∥3ρn+1-4ρn+ρn-12Δt∥2ccc+∥τn+1∥2+∥κn+1∥2+∥R1n+1∥2+∥R2n+1∥2∥2ρn-ρn-1∥2)+δ|(Zn-12(u(tn)q(tn)-UnZn)ccc-(u(tn-1)q(tn-1)-Un-1Zn-1),ξxn+1)|.
Note that
(51)(a)(3ξn+1-4ξn+ξn-12Δt,ξn+1)(a)=14Δt[(∥ξn+1∥2+∥2ξn+1-ξn∥2)(a)14Δtc-(∥ξn∥2+∥2ξn-ξn-1∥2)+∥ξn+1-2ξn+ξn-1∥2],(b)β(3ξxn+1-4ξxn+ξxn-12Δt,ξxn+1)(b)=14Δt[(∥ξxn+1∥2+∥2ξxn+1-ξxn∥2)14Δtc(b)c-(∥ξxn∥2+∥2ξxn-ξxn-1∥2)14Δtc(b)c+∥ξxn+1-2ξxn+ξxn-1∥2],(c)∥3ρn+1-4ρn+ρn-12Δt∥2≤32Δt∫tntn+1‍∥ρt∥2ds(c)∥3ρn+1-4ρn+ρn-12Δt∥2≤+12Δt∫tn-1tn‍∥ρt∥2ds,(52)|(ξxn+12(u(tn)q(tn)-UnZn)cc-(u(tn-1)q(tn-1)-Un-1Zn-1),ξxn+1)|≤|(2u(tn)(q(tn)-Zn)-u(tn)(q(tn-1)-Zn-1),ξxn+1)|+|((u(tn)-u(tn-1))(q(tn-1)-Zn-1),ξxn+1)|+|(2(u(tn)-Un)Zn-(u(tn-1)-Un-1)Zn,ξxn+1)|+|((u(tn-1)-Un-1)(Zn-Zn-1),ξxn+1)|=˙T1+T2+T3+T4.
We now estimate T1, T2, T3, and T4 as
(53)T1≤|(u(tn)(2ξn-ξn-1),ξxn+1)|+|(u(tn)(2ρn-ρn-1),ξxn+1)|≤∥u(tn)∥∞∥2ξn-ξn-1∥∥ξxn+1∥+∥u(tn)∥∞∥2ρn-ρn-1∥∥ξxn+1∥,T2=|(∫tn-1tn‍utds(ρn-1+ξn-1),ξxn+1)|≤Δt∥ut∥∞∥ρn-1+ξn-1∥∥ξxn+1∥,T3≤|(Zn(2ςn-ςn-1),ξxn+1)|+|(Zn(2ηn-ηn-1),ξxn+1)|≤∥Zn∥∞∥2ςn-ςn-1∥∥ξxn+1∥+∥Zn∥∞∥2ηn-ηn-1∥∥ξxn+1∥,T4≤|(∫tn-1tn‍Ztds(ηn-1+ςn-1),ξxn+1)|≤Δt∥Zt∥∞∥ηn-1+ςn-1∥∥ξxn+1∥.
Substituting (53) into (52), we obtain
(54)|(ξxn+12(u(tn)q(tn)-UnZn)cc-(u(tn-1)q(tn-1)-Un-1Zn-1),ξxn+1)|≤C(∥2ξn-ξn-1∥2+∥ξn-1∥2ccc+∥ρn∥2+∥ρn-1∥2+∥ςn∥2+∥ςn-1∥2ccc+∥ηn∥2+∥ηn-1∥2+∥ξxn+1∥2).
Substituting (36), (51), and (54) into (50), using (48), and summing from n=1,2,…,J, the resulting inequality becomes
(55)(1-CΔt)(∥ξJ+1∥2+∥2ξJ+1-ξJ∥2+∥ξxJ+1∥2+∥2ξxJ+1-ξxJ∥2)≤CΔt∑n=1J‍(∥ηn-1∥2+∥ρn-1∥2)+C∫0tJ+1‍∥ρt∥2ds+CΔt4(∥qtt∥L∞(L2)2+∥qttt∥L∞(L2)2c+∥uqtt∥L∞(L2)+∥utqt∥L∞(L2)+∥uttq∥L∞(L2)c+hr∥qxttt∥L∞(Hr+1)+∥qxttt∥L∞(L2)∥qtt∥L∞(L2)2)+CΔt∑n=1J‍(∥ξn∥2+∥ξxn∥2+∥2ξn-ξn-1∥2+∥2ξxn-ξxn-1∥2).
Choose Δt0 in such a way that for 0<Δt≤Δt0, (1-CΔt)>0. Then, as an application of Gronwall’s lemma, we obtain
(56)∥ξJ+1∥2+∥2ξJ+1-ξJ∥2+∥ξxJ+1∥2+∥2ξxJ+1-ξxJ∥2≤CΔt∑n=1J‍(∥ηn-1∥2+∥ρn-1∥2)+C∫0tJ+1‍∥ρt∥2ds+CΔt4(∥qtt∥L∞(L2)2+∥qttt∥L∞(L2)2+∥uqtt∥L∞(L2)+CΔt4cc+∥utqt∥L∞(L2)+∥uttq∥L∞(L2)+CΔt4cc+hr∥qxttt∥L∞(Hr+1)+∥qxttt∥L∞(L2)∥qtt∥L∞(L2)2).
Combine (28), (31), and (56) with the triangle inequality to complete the L2 and H1 error estimates for q. Furthermore, use (48) and the triangle inequality to complete the optimal error estimates for ∥u(tn)-Un∥ and ∥u(tn)-Un∥1.Remark 6.
Compared to a variety of difference methods in [17], our method is studied based on mixed element scheme (6) and (7).Remark 7.
Although some convergence proofs of multistep methods for RLW/BBM are provided in [20, 25, 29], our convergence results of multistep methods are proved based on a mixed finite element scheme. Based on the current discussion, we have to provide the detailed proofs for multistep mixed finite element methods in this paper.


## 4. Numerical Results

In order to test the viability of the proposed method, we consider a test problem. We write conservation laws as [3, 4]
(57)Q1=∫ab‍udx≃h∑j=1N‍ujn,Q2=∫ab‍(u2+μ(ux)2)dx≃h∑j=1N‍[(ujn)2+μ[(ux)jn]2],Q3=∫ab‍(u3+3u2)dx≃h∑j=1N‍[(ujn)3+3[(u)jn]2],
where Q1, Q2, and Q3 are usually called mass, momentum, and energy, respectively, which are observed to check the conservation of the numerical scheme.We consider RLW equation (1) and let, in (1), δ=γ=β=1. Then, the solitary wave solution of (1) is
(58)u(x,t)=3csech2(k[x-x0-vt]),
where
(59)k=12c1+c,v=1+c.We consider the motion of a single solitary wave and take as initial condition, withc=0.1 and x0=0,
(60)u(x,0)=0.3sech2(x211).
The corresponding exact solution with initial condition (60) is
(61)u(x,t)=0.3sech2(x-1.1t211).In this procedure, we take space-time domain as-40≤x≤60 and 0≤t≤20.In Table1, we take spatial mesh parameter h=0.125 and time discretization parameter Δt=0.1 and list the three invariants Q1, Q2, and Q3 and the optimal error estimate in L2 and L∞ norms for u at different times t=0, 4, 8, 12, 16, and 20. At the same time, we show some numerical results at time t=20 obtained by other numerical methods in Table 1. From Table 1, we find that our method is more accurate than the numerical methods in [17, 18, 28] but is less than the numerical methods in [23, 24]. From the shown results in Table 1, we can see that Q1, Q2 and Q3 keep almost constants, so the conservation for our method is very well.Table 1
Solitary wave Amp. 0.3 and the errors inL2 and L∞ norms for u, Q1, Q2, and Q3 at t=20, h=0.125, Δt=0.1, and -40≤x≤60.
Method
Time
Q
1
Q
2
Q
3
L
2 for u
L
∞ for u
Our method
0
3.9797
0.8104
2.5787
0
0
4
3.9797
0.8104
2.5786
3.6304
e
-
004
5.2892
e
-
005
8
3.9797
0.8104
2.5786
7.2873
e
-
004
5.8664
e
-
005
12
3.9797
0.8104
2.5787
1.0817
e
-
003
6.3283
e
-
005
16
3.9795
0.8104
2.5787
1.4186
e
-
003
6.8001
e
-
005
20
3.9790
0.8103
2.5785
1.7396
e
-
003
7.7154
e
-
005
[20]
20
3.9800
0.8104
2.5792
1.7569
e
-
003
6.8432
e
-
004
[21]
20
3.9820
0.8087
2.5730
4.688
e
-
003
1.755
e
-
003
[22]
20
3.9905
0.8235
2.6740
2.157
e
-
003
—
[23]
20
3.9616
0.8042
2.5583
0.018
e
-
003
1.566
e
-
003
[24]
20
3.9821
0.8112
2.5813
0.511
e
-
003
0.198
e
-
003In Tables2 and 3, we take spatial mesh parameter h=0.125 and obtain the optimal error results in L2 and L∞ norms for u at different times t=0, 4, 8, 12, 16, and 20 with different time discretization parameters Δt=0.1,0.2, and 0.4. From Tables 2 and 3, we see easily that the convergence rate for time is close to order 2. Similarly, the results for q are shown in Tables 4 and 5.Table 2
Convergence order and error inL2 norm for u of time with h=0.125 and c=0.1.
Time
Δ
t
=
0.4
Δ
t
=
0.2
Δ
t
=
0.1
Order(0.2/0.4)
Order(0.1/0.2)
4
5.3805
e
-
003
1.4267
e
-
003
3.6304
e
-
004
1.9151
1.9745
8
1.1688
e
-
002
2.9411
e
-
003
7.2873
e
-
004
1.9906
2.0129
12
1.7830
e
-
002
4.3997
e
-
003
1.0817
e
-
003
2.0188
2.0241
16
2.3751
e
-
002
5.7916
e
-
003
1.4186
e
-
003
2.0360
2.0295
20
2.9434
e
-
002
7.1148
e
-
003
1.7396
e
-
003
2.0486
2.0321Table 3
Convergence order and error inL∞ norm for u of time with h=0.125 and c=0.1.
Time
Δ
t
=
0.4
Δ
t
=
0.2
Δ
t
=
0.1
Order(0.2/0.4)
Order(0.1/0.2)
4
8.0743
e
-
004
2.0299
e
-
004
5.2892
e
-
005
1.9919
1.9403
8
9.1778
e
-
004
2.2908
e
-
004
5.8664
e
-
005
2.0023
1.9653
12
9.9866
e
-
004
2.4966
e
-
004
6.3283
e
-
005
2.0000
1.9801
16
1.0718
e
-
003
2.6685
e
-
004
6.8001
e
-
005
2.0059
1.9724
20
1.1277
e
-
003
2.8609
e
-
004
7.7154
e
-
005
1.9788
1.8907Table 4
Convergence order and error inL2 norm for q of time with h=0.125 and c=0.1.
Time
Δ
t
=
0.4
Δ
t
=
0.2
Δ
t
=
0.1
Order(0.2/0.4)
Order(0.1/0.2)
4
2.4430
e
-
003
6.4427
e
-
004
1.6260
e
-
004
1.9229
1.9863
8
5.1823
e
-
003
1.2934
e
-
003
3.1976
e
-
004
2.0024
2.0161
12
7.6616
e
-
003
1.8713
e
-
003
4.5963
e
-
004
2.0336
2.0255
16
9.8719
e
-
003
2.3787
e
-
003
5.8232
e
-
004
2.0532
2.0303
20
1.1843
e
-
002
2.8251
e
-
003
6.8991
e
-
004
2.0677
2.0338Table 5
Convergence order and error inL∞ norm for q of time with h=0.125 and c=0.1.
Time
Δ
t
=
0.4
Δ
t
=
0.2
Δ
t
=
0.1
Order(0.2/0.4)
Order(0.1/0.2)
4
1.1595
e
-
004
2.7265
e
-
005
6.3127
e
-
006
2.0884
2.1107
8
1.9712
e
-
004
4.6625
e
-
005
1.0725
e
-
005
2.0799
2.1201
12
2.5295
e
-
004
5.9837
e
-
005
1.3579
e
-
005
2.0797
2.1397
16
2.9012
e
-
004
6.8641
e
-
005
1.5594
e
-
005
2.0795
2.1381
20
3.1768
e
-
004
7.5864
e
-
005
1.7195
e
-
005
2.0661
2.1414In Tables6 and 7, the optimal error results in L2 and L∞ norms for u at different times t=0, 4, 8, 12, 16, and 20 with different spatial mesh parameters h=0.2, 0.4, and 0.8 and time discretization parameter Δt=0.01 are shown. It is easy to see that the convergence rate for space is close to order 2. The similar results for q are listed in Tables 8 and 9.Table 6
Convergence order and error inL2 norm for u of space with Δt=0.01 and c=0.1.
Time
h
=
0.8
h
=
0.4
h
=
0.2
Order(0.4/0.8)
Order(0.2/0.4)
4
1.7109
e
-
003
4.2677
e
-
004
1.1449
e
-
004
1.9940
1.8982
8
1.8945
e
-
003
4.6862
e
-
004
1.1924
e
-
004
2.0195
1.9746
12
2.1232
e
-
003
5.2130
e
-
004
1.3025
e
-
004
2.0102
2.0008
16
2.3559
e
-
003
5.7598
e
-
004
1.4421
e
-
004
2.0589
1.9978
20
2.5778
e
-
003
6.3407
e
-
004
1.7877
e
-
004
2.0358
1.8265Table 7
Convergence order and error inL∞ norm for u of space with Δt=0.01 and c=0.1.
Time
h
=
0.8
h
=
0.4
h
=
0.2
Order(0.4/0.8)
Order(0.2/0.4)
4
2.1763
e
-
003
5.8447
e
-
004
1.5502
e
-
004
1.8967
1.9147
8
4.7892
e
-
003
1.2098
e
-
003
3.0533
e
-
004
1.9850
1.9863
12
7.2371
e
-
003
1.7871
e
-
003
4.4434
e
-
004
2.0178
2.0079
16
9.4746
e
-
003
2.3084
e
-
003
5.7081
e
-
004
2.0372
2.0158
20
1.1534
e
-
002
2.7859
e
-
003
6.8980
e
-
004
2.0497
2.0139Table 8
Convergence order and error inL2 norm for q of space with Δt=0.01 and c=0.1.
Time
h
=
0.8
h
=
0.4
h
=
0.2
Order(0.4/0.8)
Order(0.2/0.4)
4
2.3426
e
-
004
5.4759
e
-
005
1.2581
e
-
005
2.0969
2.1218
8
4.2831
e
-
004
1.0038
e
-
004
2.2853
e
-
005
2.0932
2.1350
12
5.7270
e
-
004
1.3452
e
-
004
3.0443
e
-
005
2.0900
2.1436
16
6.7812
e
-
004
1.5951
e
-
004
3.5927
e
-
005
2.0879
2.1505
20
7.5693
e
-
004
1.7832
e
-
004
4.0176
e
-
005
2.0857
2.1501Table 9
Convergence order and error inL∞ norm for q of space with Δt=0.01 and c=0.1.
Time
h
=
0.8
h
=
0.4
h
=
0.2
Order(0.4/0.8)
Order(0.2/0.4)
4
1.1957
e
-
003
3.1367
e
-
004
7.8812
e
-
005
1.9305
1.9928
8
2.4344
e
-
003
6.0206
e
-
004
1.4825
e
-
004
2.0156
2.0219
12
3.4768
e
-
003
8.3962
e
-
004
2.0540
e
-
004
2.0500
2.0313
16
4.3743
e
-
003
1.0402
e
-
003
2.5371
e
-
004
2.0722
2.0356
20
5.1677
e
-
003
1.2153
e
-
003
2.9544
e
-
004
2.0882
2.0404Figure1 shows the surface for the exact solution u in space-time domain ((x,t)∈[-20,60]×[0,20]), and the corresponding surface for the numerical solution uh with h=2 and Δt=0.4 is described in Figure 2. From Figures 1 and 2, we see easily that the exact solution u is approximated very well by the numerical solution uh. In Figures 3 and 4, we show the surface for the exact solution q and the numerical solution qh, respectively, with h=2 and Δt=0.4 and obtain a good approximation solution qh for the exact solution q.Figure 1
Surface for exact solutionu.Figure 2
Surface for numerical solutionuh.Figure 3
Surface for exact solutionq.Figure 4
Surface for numerical solutionqh.The comparison between the exact solutionu and the numerical solution uh is described at different times t=4,8,12,16, and 20 with h=0.125 and Δt=0.2 in Figure 5. The similar comparison for the exact solution q and the numerical solution qh is shown in Figure 6. Figures 5 and 6 show that the solitary wave for u and q moves to the right with unchanged form and velocity, respectively. Furthermore, the exact solutions u and q are approximated well by the numerical solutions uh and qh, respectively.Figure 5
Comparison betweenu and uh at times t=4, 8, 12, 16, and 20 with h=0.125 and Δt=0.2.Figure 6
Comparison betweenq and qh at times t=48, 12, 16, and 20 with h=0.125 and Δt=0.2.In Figure7, we show the comparison between u and uh at different spaces x=0,10,20, and 30 with h=0.125 and Δt=0.2 to verify the efficiency for the proposed scheme in this paper. Figure 8 describes a similar result for q.Figure 7
Comparison betweenu and uh at spaces x=0, 10, 20, and 30 with h=0.125 and Δt=0.2.Figure 8
Comparison betweenq and qh at spaces x=0, 10, 20, and 30 with h=0.125 and Δt=0.2.From the previous analysis in Tables1–9 and Figures 1–8, we can see that the numerical results confirm the theoretical results of Theorem 5 and our method is efficient for RLW equation.


## 5. Concluding Remarks

In this paper, we propose and analyze an explicit multistep mixed finite element method, which combines spatial mixed finite element method and time explicit multistep method, for RLW equation. We discuss the numerical process for our method, prove the theoretical results for the fully discrete explicit multistep mixed scheme, obtain the optimal convergence order, and compare our method’s accuracy with some other numerical schemes. Compared with the numerical method in [20, 25, 26], our method can obtain the optimal error estimates in L2 and H1 norms for the scalar unknown u and its flux q=ux simultaneously. From our numerical results, we can see that our method is efficient for RLW equation.



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*Source: 768976-2013-05-22.xml*