# A New Numerical Approximation Method for Two-Dimensional Wave Equation with Neumann Damped Boundary

Authors: Jiankang Liu; Suying Zhang
Journal: Complexity
 (2020)

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## Abstract

In this paper, a fully discretized finite difference scheme is derived for two-dimensional wave equation with damped Neumann boundary condition. By discrete energy method, the proposed difference scheme is proven to be of second-order convergence and of unconditional stability with respect to both initial conditions and right-hand term in a proper discretizedL2 norm. The theoretical result is verified by a numerical experiment.


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## Body

## 1. Introduction

In this paper, we consider the finite difference discretization for the following initial boundary value problem (IBVP) of wave equation in a squareΩ=0,1×0,1:(1)wttx,y,t−Δwx,y,t=fx,y,t,x,y,t∈Ω×0,T,(2)wx,y,0=φx,y,wtx,y,0=ψx,y,x,y∈Ω,(3)w0,y,t=0,wx1,y,t=−wt1,y,t,y∈0,1,t∈0,T,(4)wx,0,t=0,wyx,1,t=−wtx,1,t,x∈0,1,t∈0,T,where φx,y, ψx,y, and fx,y,t are given sufficient smooth functions, and φ0,y=φx,0=0, ψ0,y=ψx,0=0, φx1,y+ψ1,y=φyx,1+ψx,1=0, which means that the initial conditions are compatible with boundary conditions.This system arises in many important models for distributed parameter control systems. In particular, in the model of a vibrating flexible membrane, the solutionw represents the transverse displacement of the membrane, and in models for acoustic pressure fields, the solution w represents the fluid pressure (see, [1, 2] for more examples). The boundary conditions on the right and top sides of Ω in (3) and (4) are called Neumann actuations with damped controller from the viewpoint of control theory. Here, this kind of boundary condition is regarded as damped Neumann boundary. For this special boundary condition, it is more complicated to construct a proper energy function to prove a priori estimate of proposed finite difference scheme than that of classical boundary conditions (Dirichlet, Neumann, or Robin).The numerical analysis of second-order hyperbolic equations has been extensively studied for decades. Many numerical methods have been used to approximate this kind of problem, including Finite Element Methods [3, 4], Finite Volume Methods [5, 6], Finite Difference Methods [7, 8], Discontinuous Galerkin methods [9–11], Mixed Finite Element Methods [12] and so on. In [4], wave equation with homogeneous boundary condition is considered, in which a posteriori error bounds in the L∞L2 norm for finite element methods is derived under minimal regularity assumptions. A finite volume scheme on the nonconforming meshes for multidimensional wave equation is constructed in [6], where the error estimates for the approximations of the exact solution and its first derivatives is derived. An unconditionally stable and second-order convergent finite difference scheme is constructed for one-dimensional linear hyperbolic equation in [8]. As far as hyperbolic conservation law is concerned, there are many distinguished papers focused on this topic, for example, the classical ENO/WENO schemes [13], the improved ENO/WENO schemes [14–17], and so on. All linear numerical schemes are either dispersive or dissipative. The computational dispersion can lead to noise in the numerical solution. Dispersion and dissipation phenomena were investigated in [18–22], where some composite schemes were proposed to reduce the dispersive effect on the numerical solution. High order compact finite difference scheme is an efficient way for solving PDEs. In [23–25], the high order compact finite difference scheme was constructed for hyperbolic equations subject to homogeneous boundary condition.There are also some investigations of the numerical methods for viscous or strongly damped hyperbolic equation [26–29]. However, most of them just considered the homogeneous boundary problems. In [3], the generalized version of this kind of boundary condition in [3, (2.2b)]) was considered in the context of finite element methods. In [30], the mixed finite element formulation for second-order hyperbolic equation with absorbing boundary condition was investigated. Recently, the weak formulation of hyperbolic problems with inhomogeneous Dirichlet and Neumann boundary was considered in [31]. To the best of our knowledge, there is no contribution to the finite difference approximation to the 2D wave equation with damped Neumann boundary condition so far.The rest of this paper is organized as follows. Some basic notations and lemmas are given in Section2, which is essential to the analysis of finite difference schemes. Section 3 constructs the finite difference scheme for (1)–(4). A priori estimate of the proposed difference scheme is shown in Section 4. The unique solvability, convergence, and stability of proposed finite difference scheme are proved in Section 5. A numerical experiment is conducted in Section 6, before a conclusion is stated in Section 7.


## 2. Preliminary

Letm and n be two positive integers and assume that the space step and time step are h=1/m and τ=T/n, respectively. Define Ωh=xi,yjxi=ih,yj=jh,0≤i,j≤m, Ωτ=tktk=kτ,0≤k≤n and Ωhτ=Ωh×Ωτ. Let w=wi,jk0≤i,j≤m,0≤k≤n be the grid function on Ωhτ. We introduce the following difference and averaging operators:(5)wijk+1/2=wijk+wijk+12,δtwijk+1/2=wijk+1−wijkτ,Dt^wijk=wijk+1−wijk−12τ,δt2wijk=δtwijk+1/2−δtwijk−1/2τ,δxwi+1/2,jk=wi+1,jk−wi,jkh,δywi,j+1/2k=wi,j+1k−wi,jkh,δx2wijk=δxwi+1/2,jk−δxwi−1/2,jkh,δy2wijk=δywi,j+1/2k−δywi,j−1/2kh,wijk¯=wijk+1+wijk−12.The following lemmas (Lemma1–Lemma 4) are necessary for analyzing the truncation error of the difference scheme and for proving the convergence of the difference scheme, which are analogous to lemmas in [32].Lemma 1.
Leth>0 and c be two constants.(a)
Ifgx∈C2c−h,c+h, then(6)gc=12gc−h+gc+h−h22g″ξ0,c−h<ξ0<c+h.(b)
Ifgx∈C2c,c+h, then(7)g′c=1hgc+h−gc−h2g″ξ1,c−h<ξ1<c+h.(c)
Ifgx∈C2c−h,c, then(8)g′c=1hgc−gc−h+h2g″ξ2,c−h<ξ2<c.(d)
Ifgx∈C3c−h,c+h, then(9)g′c=12hgc+h−gc−h−h26g‴ξ3,c−h<ξ3<c+h.(e)
Ifgx∈C4c−h,c+h, then(10)g″c=1h2gc+h−2gc+gc−h−h212g4ξ4,c−h<ξ4<c+h.(f)
Ifgx∈C3c−h,c, then(11)g″c=2hg′c−gc−gc−hh+h3g‴ξ5,c−h<ξ5<c.Lemma 2. (Gronwall Inequality).
SupposeFk,Gkk≥0 are nonnegative sequences such that Fk+1≤1+cτFk+τGk,k=0,1,2,…, then(12)Fk≤expckτF0+τ∑l=1k−1Gl,k=0,1,2,…,where c is a nonnegative constant.Lemma 3.
Suppose the mesh grids beIh=xi0≤i≤m, where xi=ih, h=1/m. Let u=ui0≤i≤m be mesh function on Ih such that u0=0, then(13)u∞≤u1,u≤22u1,where u1=h∑i=1mδxui−1/22, u=h1/2u02+∑i=1m−1ui2+1/2um2, u∞=max0≤i≤mui.Proof.
For1≤i≤m, we have(14)ui=∑j=1iuj−uj−1=h∑j=1iδxuj−1/2.
Square both sides of (14) and apply the Cauchy–Schwarz inequality, then(15)ui2≤h∑j=1i12h∑j=1iδxuj−1/22=xih∑j=1iδxuj−1/22≤h∑j=1mδxuj−1/22,which directly implies(16)u∞≤u1.
The first inequality of (15) implies(17)ui2≤xih∑j=1iδxuj−1/22≤xiu12.
Multiply both sides of (17) by h, and sum for i from 1 to m−1, then(18)h∑i=1m−1ui2≤h∑i=1m−1xiu12=12mm−1h2u12.
Thus, we have(19)u2=h∑i=1m−1ui2+h2um2≤h2mm−1h+h2u12=12u12,which implies(20)u≤22u1,and ends the proof.Introduce a grid function space onΩh:(21)V=vv=viji,j=0m;vi,0=0,0≤i≤m;v0,j=0,1≤j≤m.Then forv∈V, define the norms(22)v=h2∑i,j=1m−1vij2+h212∑j=1m−1vmj2+h212∑i=1m−1vim2+h214vmm2,δxv=h2∑i=0m−112δxvi+1/2,02+∑j=1m−1δxvi+1/2,j2+12δxvi+1/2,m2,δyv=h2∑j=0m−112δyv0,j+1/22+∑i=1m−1δyvi,j+1/22+12δyvm,j+1/22.Lemma 4.
Letv=vij0≤i,j≤m be mesh function on Ωh such that v0j=vi0=0 for 1≤i,j≤m, then we have(23)v≤12v1,where v1=δxv2+δyv21/2.Proof.
From Lemma3, we have(24)h∑i=1m−1uij2+h2umj2≤12h∑i=0m−1δxui+1/2,j2,1≤j≤m.
Multiplying both sides of (24) by h and summing for j from 1 to m−1, we get(25)h2∑i,j=1m−1uij2+h22∑j=1m−1umj2≤12h2∑i=0m−1∑j=1m−1δxui+1/2,j2.
Multiplying both sides of (24) by h/2 and letting j=m in (24), then after combining it with (25), we get(26)u2≤12δxu2.
Similarly, we have(27)u2≤12δyu2.
Combining (26) and (27), it follows(28)u≤12u1.
This completes the proof.


## 3. Derivation of the Finite Difference Scheme

Define the grid functionW=Wijk0≤i,j≤m,0≤k≤n by(29)Wijk=wxi,yj,tk,0≤i,j≤m,0≤k≤n.Consider equation (1) at the inner points xi,yj,tk in Ωhτ, then it follows(30)∂2wxi,yj,tk∂t2−∂2wxi,yj,tk∂x2−∂2wxi,yj,tk∂y2=fxi,yj,tk,1≤i,j≤m−1,1≤k≤n−1.According to Lemma1(e), it follows(31)δt2Wijk−δx2Wijk¯−δy2Wijk¯=fxi,yj,tk+Rijk,1≤i,j≤m−1,1≤k≤n−1,where Rijk=Oτ2+h2. These are discrete equations at inner points.Next we will pay our main attention to the derivation of difference equations at the boundary points. Considering the boundary condition 1.3 atxm,yj,tk, it follows(32)∂wxm,yj,tk∂t+∂wxm,yj,tk∂x=0,1≤j≤m−1.Similarly, the boundary condition 1.4 atxi,ym,tk satisfies(33)∂wxi,ym,tk∂t+∂wxi,ym,tk∂y=0,1≤i≤m−1.By (3) and (4), the differential equation at the corner point xm,ym,tk reads(34)2∂wxm,ym,tk∂t+∂wxm,ym,tk∂x+∂wxm,ym,tk∂y=0.According to Lemma1 (a), (c), and (d), we have(35)∂wxm,yj,tk∂t=Wxm,yj,tk+1−Wxm,yj,tk−12τ+Oτ2=Dt^Wmjk+Oτ2,(36)∂wxi,ym,tk∂t=Wxi,ym,tk+1−Wxi,ym,tk−12τ+Oτ2=Dt^Wimk+Oτ2,(37)∂wxm,yj,tk∂x=12∂wxm,yj,tk−1∂x+∂wxm,yj,tk+1∂x+Oτ2=Dx−Wmjk¯+h2δt2Wmjk−δy2Wmjk¯−fxm,yj,tk+Oτ2+h2,(38)∂wxi,ym,tk∂y=12∂wxi,ym,tk−1∂y+∂wxi,ym,tk+1∂y+Oτ2=Dy−Wimk¯+h2δt2Wimk−δx2Wimk¯−fxi,ym,tk+Oτ2+h2.Applying (35) and (37) into (32), we get(39)Dx−Wmjk¯+Dt^Wmjk+h2δt2Wmjk−h2δy2Wmjk¯=h2fxm,yj,tk+Rmjk,1≤j≤m−1,1≤k≤n−1,where Rmjk=Oτ2+h2. Similarly, we have(40)Dy−Wimk¯+Dt^Wimk+h2δt2Wimk−h2δx2Wimk¯=h2fxi,ym,tk+Rimk,(41)Dx−Wmmk¯+Dy−Wmmk¯+2Dt^Wmmk+h2δt2Wmmk=hfxm,ym,tk+Rmmk,1≤i≤m−1,1≤k≤n−1,where Rimk=Oτ2+h2 and Rmmk=Oτ2+h2.Whenk=0, equation (1) at initial grids xi,yj,t0 satisfies(42)∂2wxi,yj,t0∂t2−∂2wxi,yj,t0∂x2−∂2wxi,yj,t0∂y2=fxi,yj,t0,1≤i,j≤m−1.From Lemma1(f) we have,(43)∂2wxi,yj,t0∂t2=2τδtWij1/2−wtxi,yj,t0+Oτ.Putting (43) into (42), then by Lemma 1(a) and (e), the following equations hold:(44)2τδtWij1/2−wtxi,yj,t0−δx2Wij1/2−δy2Wij1/2=fxi,yj,t0+rij,1≤i,j≤m−1,where rij=Oτ+h2. At xm,yj,t1/2, we construct the following difference scheme:(45)δxWm−1/2,j1/2+δtWmj1/2+hτδtWmj1/2−h2δy2Wmj1/2=hτψxm,yj+h2fxm,yj,t0+rmj,where rmj=Oτ2+h2. Similarly, at xi,ym,t1/2 and xm,ym,t1/2, we have(46)δyWi,m−1/21/2+δtWim1/2+hτδtWim1/2−h2δx2Wim1/2=hτψxi,ym+h2fxi,ym,t0+rim,(47)δxWm−1/2,m1/2+δyWm,m−1/21/2+2δtWmm1/2+hτδtWmm1/2=2hτψxm,ym+hfxm,ym,t0+rmm,respectively, where rim=Oτ2+h2 and rmm=Oτ2+h2. For the initial condition and the left and bottom boundaries, we have(48)Wij0=φxi,yj,0≤i,j≤m,(49)W0jk=Wi0k=0,0≤i≤m,1≤j≤m,1≤k≤n.Finally, by dropping the infinitesimals in (31), (39)–(47), with Wijk replaced by wijk, we get the finite difference schemes for (1)–(4) as follows(50)δt2wijk−δx2wijk¯−δy2wijk¯=fxi,yj,tk,1≤i,j≤m−1,1≤k≤n−1,(51)Dx−wmjk¯+Dt^wmjk+h2δt2wmjk−h2δy2wmjk¯=h2fxm,yj,tk,1≤j≤m−1,1≤k≤n−1,(52)Dy−wimk¯+Dt^wimk+h2δt2wimk−h2δx2wimk¯=h2fxi,ym,tk,1≤j≤m−1,1≤k≤n−1,(53)Dx−wmmk¯+Dy−wmmk¯+2Dt^wmmk+h2δt2wmmk=h2fxm,ym,tk,1≤k≤n−1,(54)2τδtwij1/2−δx2wij1/2−δy2wij1/2=fxi,yj,t0+2τψxi,yj,1≤i,j≤m−1,(55)δxwm−1/2,j1/2+δtwmj1/2+hτδtwmj1/2−h2δy2wmj1/2=hτψxm,yj+h2fxm,yj,t0,1≤j≤m−1,(56)δywi,m−1/21/2+δtwim1/2+hτδtwim1/2−h2δx2wim1/2=hτψxi,ym+h2fxi,ym,t0,1≤i≤m−1,(57)δxwm−1/2,m1/2+δywm,m−1/21/2+2δtwmm1/2+hτδtwmm1/2=2hτψxm,ym+h2fxm,ym,t0,(58)wij0=φxi,yj,0≤i,j≤m,(59)wi0k=0,w0jk=0,0≤i≤m,1≤j≤m,1≤k≤n.


## 4. A Priori Estimate of the Difference Scheme

In order to prove the convergence and stability of difference schemes, we give a priori estimate of difference schemes (50)–(59).Theorem 1.
Suppose thatuijk0≤i,j≤m,0≤k≤n solves the following difference scheme:(60)δt2uijk−δx2uijk¯−δy2uijk¯=fijk,1≤i,j≤m−1,1≤k≤n−1,(61)Dx−umjk¯+Dt^umjk+h2δt2umjk−h2δy2umjk¯=fmjk,1≤j≤m−1,1≤k≤n−1,(62)Dy−uimk¯+Dt^uimk+h2δt2uimk−h2δx2uimk¯=fimk,1≤i≤m−1,1≤k≤n−1,(63)Dx−ummk¯+Dy−ummk¯+2Dt^ummk+h2δt2ummk=fmmk,1≤k≤n−1,(64)2τδtuij1/2−δx2uij1/2−δy2uij1/2=fij0,1≤i,j≤m−1,(65)δxum−1/2,j1/2+δtumj1/2+hτδtumj1/2−h2δy2umj1/2=fmj0,1≤j≤m−1,(66)δyui,m−1/21/2+δtuim1/2+hτδtuim1/2−h2δx2uim1/2=fim0,1≤i≤m−1,(67)δxum−1/2,m1/2+δyum,m−1/21/2+2δtumm1/2+hτδtumm1/2=fmm0,(68)uij0=φxi,yj,0≤i,j≤m,(69)ui0k=0,u0jk=0,0≤i≤m,1≤j≤m,1≤k≤n,then for arbitrary grid ratio λ=τ/h, we have(70)δtuk+1/22+12uk+112+uk12≤exp32Tφ12+τ24f02+hτ4∑j=1m−1fmj02+hτ4∑i=1m−1fim02+hτ16fmm02+3τ2∑l=1kGl,1≤k≤n−1,where(71)Gl=h2∑i,j=1m−1fijk2+h2∑j=1m−1fmjl2+h2∑i=1m−1fiml2+h8fmml2,1≤l≤n−1.Proof.
Define(72)Ek=δtuk+1/22+12uk+112+uk12.
Multiplying both sides of (60) by 2h2Dt^uijk, and taking summation for i and j from 1 to m−1, we get(73)2h2∑i,j=1m−1δt2uijkDt^uijk−2h2∑i,j=1m−1δx2uijk¯Dt^uijk−2h2∑i,j=1m−1δy2uijk¯Dt^uijk=2h2∑i,j=1m−1fijkDt^uijk,1≤k≤n−1.
The first term in (73) implies(74)2h2∑i,j=1m−1δt2ui,jkDt^uijk=1τh2∑i,j=1m−1δtuijk+1/22−h2∑i,j=1m−1δtuijk−1/22.
The second term and the third term in (73) imply(75)−2h2∑i,j=1m−1δx2uijk¯Dt^uijk=−2h∑i,j=1m−1δxui+1/2,jk¯−δxui−1/2,jk¯Dt^uijk=2h2∑i=0m−1∑j=1m−1δxui+1/2,jk¯Dt^δxui+1/2,jk−2h∑j=1m−1Dx−umjk¯Dt^umjk=12τh2∑i=0m−1∑j=1m−1δxui+1/2,jk+12−h2∑i=0m−1∑j=1m−1δxui+1/2,jk−12−2h∑j=1m−1Dx−umjk¯Dt^umjk,(76)−2h2∑i,j=1m−1δy2uijk¯Dt^uijk=12τh2∑i=1m−1∑j=0m−1δyui,j+1/2k+12−h2∑i=1m−1∑j=0m−1δyui,j+1/2k−12−2h∑i=1m−1Dy−uimk¯Dt^uimk,respectively. Putting (74), (75) and (76) into (73), we get(77)1τh2∑i,j=1m−1δtuijk+1/22−h2∑i,j=1m−1δtuijk−1/22+12τh2∑i=0m−1∑j=1m−1δxui+1/2,jk+12−h2∑i=0m−1∑j=1m−1δxui+1/2,jk−12+12τh2∑i=1m−1∑j=0m−1δyui,j+1/2k+12−h2∑i=1m−1∑j=0m−1δyui,j+1/2k−12=2h2∑i,j=1m−1fijkDt^uijk+2h∑j=1m−1Dx−umjk¯Dt^umjk+2h∑i=1m−1Dy−uimk¯Dt^uimk,where(78)2h2∑i,j=1m−1fijkDt^uijk≤h2∑i,j=1m−1Dt^uijk2+h2∑i,j=1m−1fijk2≤12h2∑i,j=1m−1δtuijk+1/22+h2∑i,j=1m−1δtuijk−1/22+h2∑i,j=1m−1fijk2.
Considering (61)–(63) and by inequality 2b−aa≤1/2b2, we get(79)2h∑j=1m−1Dx−umjk¯Dt^umjk+2h∑i=1m−1Dy−uimk¯Dt^uimk=h∑j=1m−12fmjk−Dt^umjkDt^umjk−hδt2umjkDt^umjk+hδy2umjk¯Dt^umjk+h∑i=1m−12fimk−Dt^uimkDt^uimk−hδt2uimkDt^uimk+hδx2uimk¯Dt^uimk≤−h22τ∑j=1m−1δtumjk+1/22−δtumjk−1/22−h24τ∑j=0m−1δyum,j+1/2k+12−δyum,j+1/2k−12−h22τ∑i=1m−1δtuimk+1/22−δtuimk−1/22−h24τ∑i=0m−1δxui+1/2,mk+12−δxui+1/2,mk−12+h8fmmk2−h24τδtummk+1/22−δtummk−1/22+h2∑j=1m−1fmjk2+h2∑i=1m−1fimk2.
According to the definition ofEk, (77) implies(80)1τEk−Ek−1≤12Ek+Ek−1+h2∑i,j=1m−1fijk2+h2∑j=1m−1fmjk2+h2∑i=1m−1fimk2+h8fmmk2,1≤k≤n−1.
If we confineτ≤2/3, then(81)Ek≤1+3τ2Ek−1+3τ2Gk,1≤k≤n−1.
By Gronwall inequality in Lemma2, we get(82)Ek≤exp32kτE0+3τ2∑l=1kGl,1≤k≤n−1.
Next we estimateE0. Multiplying (64) by h2δtuij1/2, and summing up for i and j from 1 to m−1, we have(83)2τh2∑i,j=1m−1δtuij1/22−h2∑i,j=1m−1δx2uij1/2+δy2uij1/2δtuij1/2=h2∑i,j=1m−1fij0δtuij1/2,which similarly implies(84)2τδtu1/22+12τu112−u012≤1τδtu1/22+τ4f02+h4∑j=1m−1fmj02+h4∑i=1m−1fim02+h16fmm02.
According to the definition ofE0, we obtain(85)E0=δtu1/22+12u112+u012≤u012+τ24f02+hτ4∑j=1m−1fmj02+hτ4∑i=1m−1fim02+hτ16fmm02.
Putting (85) into (82) and noticing kτ≤T, we achieve (70). This completes the proof. □


## 5. Existence, Convergence, and Stability

In this section, we will discuss the unique solvability, convergence, and stability of the difference scheme (50)–(59).Theorem 2.
The difference scheme (50)–(59) is uniquely solvable.Proof.
Notice thatw0 is known by the initial condition. The difference scheme for w1 is (54)–(59). We consider the following homogeneous equations of (54)–(59):(86)2τ2wij1−12δx2wij1−12δy2wij1=0,1≤i,j≤m−1,(87)12δxwm−1/2,j1+1τwmj1+hτ2wmj1−h4δy2wmj1=0,1≤j≤m−1,(88)12δywi,m−1/21+1τwim1+hτ2wim1−h4δx2wim1=0,1≤i≤m−1,(89)12δxwm−1/2,m1+12δywm,m−1/21+2τwmm1+hτ2wmm1=0,(90)wi01=0,0≤i≤m,w0j1=0,1≤j≤m.
Multiplying (86) by h2wij1, summing up for i and j from 1 to m−1, and combining with (87)–(90), we get(91)2τ2w12+12w112+hτ∑j=1m−1wmj12+∑i=1m−1wim12+wmm12=0,which implies wij1=00≤i,j≤m. Thus, w1 is uniquely solvable.
Now suppose thatwk−1 and wk are uniquely solvable. The difference schemes for wk+1 satisfy (50)–(53) and (59). The homogeneous equations read:(92)1τ2wijk+1−12δx2wijk+1−12δy2wijk+1=0,1≤i,j≤m−1,1≤k≤n−1,(93)12Dx−wmjk+1+12τwmjk+1+h2τ2wmjk+1−h4δy2wmjk+1=0,1≤j≤m−1,1≤k≤n−1,(94)12Dy−wimk+1+12τwimk+1+h2τ2wimk+1−h4δx2wimk+1=0,1≤i≤m−1,1≤k≤n−1,(95)12Dx−wmmk+1+12Dy−wmmk+1+1τwmmk+1+h2τ2wmmk+1=0,1≤k≤n−1,(96)wi0k=0,w0jk=0,0≤i≤m,0≤j≤m,1≤k≤n.
Multiplying (92) by h2wijk+1, and summing up for i and j from 1 to m−1, and applying equations (93)–(96), we have(97)1τ2wk+12+12wk+112+h2τ∑j=1m−1wmjk+12+∑i=1m−1wimk+12+wmmk+12=0,which means that wijk+1=00≤i,j≤m. Thus wk+1 is uniquely solvable.
According to the principle of induction, the difference scheme of (50)–(59) is uniquely solvable for all wk1≤k≤n. This completes the proof.Define the pointwise error by(98)eijk=Wijk−wijk,0≤i,j≤m,0≤k≤n.The following theorem states the convergence of difference scheme (50)–(59).Theorem 3.
Suppose thatwx,y,t∈Cx,y,t4,4,3Ω¯×0,T is the solution of (1)–(4) and wijk0≤i,j≤m,0≤k≤n is the solution of difference scheme (50)–(59). Then there exists a constant CT such that(99)δtek+1/22+12ek+112+ek12≤CTτ2+h22,0≤k≤n−1,where CT=exp3/2T5/4+3/2Tc2.Proof.
Subtracting (50)–(59) from (31), (39)–(41) and (44)–(49) respectively, we have the following error equation(100)δt2eijk−δx2eijk¯−δy2eijk¯=Rijk,1≤i,j≤m−1,1≤k≤n−1,Dx−emjk¯+Dt^emjk+h2δt2emjk−h2δy2emjk¯=Rmjk,1≤j≤m−1,1≤k≤n−1,Dy−eimk¯+Dt^eimk+h2δt2eimk−h2δx2eimk¯=Rimk,1≤i≤m−1,1≤k≤n−1,Dx−emmk¯+Dy−emmk¯+2Dt^emmk+h2δt2emmk=Rmmk,1≤k≤n−1,2τδteij1/2−δx2eij1/2−δy2eij1/2=rij,1≤i,j≤m−1,δxem−1/2,j1/2+δtemj1/2+hτδtemj1/2−h2δy2emj1/2=rmj,1≤j≤m−1,δyei,m−1/21/2+δteim1/2+hτδteim1/2−h2δx2eim1/2=rim,1≤i≤m−1,δxem−1/2,m1/2+δyem,m−1/21/2+2δtemm1/2+hτδtemm1/2=rmm,eij0=0,0≤i,j≤m,e0jk=ei0k=0,0≤i≤m,1≤j≤m,0≤k≤n.
According to Theorem1, we get(101)δtek+1/22+12ek+112+ek12≤exp32kτe012+τ24r2+hτ4∑j=1m−1rmj2+hτ4∑i=1m−1rim2+hτ16rmm2+3τ2∑l=1kG^l,1≤k≤n−1,where(102)G^l=Rl2+h2∑j=1m−1Rmjl2+h2∑i=1m−1Riml2+h8Rmml2,1≤l≤n−1.
Taking into account the order ofrij and Rijk0≤i,j≤m,1≤k≤n, and the fact that eij0=0, there exists a generic constant c such that(103)δtek+1/22+12ek+112+ek12≤exp32kττ24c2τ+h22+τh4c2τ2+h22+τh4c2τ2+h22+τh16c2τ2+h22+32kτc2τ2+h22≤exp32T54+32Tc2τ2+h22=CTτ2+h22,0≤k≤n−1.
This completes the proof.According to Lemma4, we can conclude that ek=Oτ2+h21≤k≤n, namely, the difference solution is of second-order convergence in L2 norm.Theorem 4.
For any fixed grid ratioλ=τ/h, the difference scheme (50)–(59) is stable with respect to the initial value φ, ψ, and the source term f. More precisely, we have(104)δtwk+1/22+12wk+112+wk12≤exp32Tφ12+cψ2+c9f02+3τ2∑l=1kfl2,0≤k≤n−1,where c=max2,1/λ.Proof.
Suppose that the solution of difference scheme (50)–(59) is wijk0≤i,j≤m,0≤k≤n, then for any λ we can get similarly to Theorem 1(105)δtwk+1/22+12wk+112+wk12≤exp32TE0+3τ2∑l=1kfl2,0≤k≤n−1,where(106)E0=δtw1/22+12w112+w012≤φ12+2h2∑i,j=1m−1ψij2+h32τ∑j=1m−1ψmj2+h32τ∑i=1m−1ψim2+h38τψmm2+τ2h22∑i,j=1m−1fij02+τh38∑j=1m−1fmj02+τh38∑i=1m−1fim02+τh332fmm02.
Taking account into the fact thatτ≤2/3, h=τ/λ≤2/3λ, there exists constant c=max2,1/λ such that(107)E0≤φ12+cψ2+c9f02,which ends the proof.


## 6. Numerical Result

In order to test the convergence order of proposed difference schemes (50)–(59), we consider a two-dimensional damped wave equation, in which the source function fx,y,t is defined by(108)fx,y,t=243π216exp−94πtsin94πxsin94πy,with initial conditions φx,y and ψx,y designed as(109)φx,y=sin94πxsin94πy,ψx,y=−94πsin94πxsin94πy.For this specified problem, the exact solution is(110)wx,y,t=exp−94πtsin94πxsin94πy.For the source term and initial conditions defined above, the evolution mesh and contour map of the numerical solution of difference schemes (50)–(59) with N = 40 and T = 0.5 are shown in Figure 1.Figure 1
The evolution mesh (a) and contour map (b) of numerical solution.
(a)(b)Let us denote the error by(111)EWτ,h=wτ,h−Wτ,h.Table1 provides the computed results with a fixed h=1/100. The data demonstrate that the temporal convergence order of difference scheme (50)–(59) is second order in discrete L2–norm, which is in accord with theoretical analysis of Theorem 3. When the temporal size is fixed with τ=1/200, Table 2 gives the discrete L2–errors and numerical convergence orders in space direction. From these data, we can see that the spatial convergence order of difference scheme (50)–(59) is also second order, which is in agreement with theoretical results in Theorem 3.Table 1
Numerical convergence of scheme (50)–(59) in time (h=1/100, T=1).
τEWτ,hOrder1/201.2948e − 002—1/402.9438e − 0032.13701/501.8099e − 0032.17991/601.2057e − 0032.22801/708.4660e − 0042.2938Table 2
Numerical convergence of scheme (50)–(59) in space (τ=1/200, T=1).
hEWτ,hOrder1/202.4487e − 003—1/301.0292e − 0032.13771/405.4095e − 0042.23581/503.1889e − 0042.36831/602.0234e − 0042.4950The convergence results with fixed grid ratioλ=1 and λ=2 are shown in Tables 3 and 4, respectively. For these two cases, the proposed difference scheme (50)–(59) can preserve both the convergence and stability. From the numerical results, we can verify the unconditional stability of difference scheme (50)–(59). Also for other different values of λ, the same stability results can be drawn, which are accordant with Theorem 4. The curve of convergence order in Tables 1–4 is shown in Figure 2.Table 3
Numerical convergence of scheme (50)–(59) with λ=1 (T=1).
hEWτ,hOrder1/201.1049e − 002—1/304.8033e − 0032.05451/402.6446e − 0032.07451/501.6725e − 0032.04621/601.1536e − 0032.0461Table 4
Numerical convergence of scheme (50)–(59) with λ=2 (T=1).
hEWτ,hOrder1/302.1489e − 002—1/401.2564e − 0021.86561/507.9166e − 0032.06991/605.4036e − 0032.09461/703.9183e − 0032.0850Figure 2
The curve of convergence order in Tables1–4.


## 7. Conclusion

In this paper, a finite difference scheme is constructed for the two-dimensional wave equation with a special boundary condition, damped boundary condition. By introducing a proper discreteL2–norm, the proposed finite difference scheme is proved to be second-order convergent in both time and space and unconditionally stable with respect to both initial conditions and source term. The theoretical analysis method can be applied to three-dimensional problems accordingly.



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