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

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 discretized L2 norm. -e theoretical result is verified by a numerical experiment.


Introduction
In this paper, we consider the finite difference discretization for the following initial boundary value problem (IBVP) of wave equation in a square Ω � (0, 1) × (0, 1): w tt (x, y, t) − Δw(x, y, t) � f(x, y, t), (x, y, t) ∈ Ω ×(0, T], (1) w(x, y, 0) � φ(x, y), w t (x, y, 0) � ψ(x, y), w(0, y, t) � 0, w(x, 0, t) � 0, w y (x, 1, t) � − w t (x, 1, t), where φ(x, y), ψ(x, y), and f(x, y, t) are given sufficient smooth functions, and φ(0, y) � φ(x, 0) � 0, ψ(0, y) � ψ(x, 0) � 0, φ x (1, y) + ψ(1, y) � φ y (x, 1) + ψ(x, 1) � 0, which means that the initial conditions are compatible with boundary conditions. is system arises in many important models for distributed parameter control systems. In particular, in the model of a vibrating flexible membrane, the solution w 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). e 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). e 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][10][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 ∞ (L 2 ) 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 secondorder 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][15][16][17], and so on. All linear numerical schemes are either dispersive or dissipative. e computational dispersion can lead to noise in the numerical solution. Dispersion and dissipation phenomena were investigated in [18][19][20][21][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][24][25], the high order compact finite difference scheme was constructed for hyperbolic equations subject to homogeneous boundary condition.
ere are also some investigations of the numerical methods for viscous or strongly damped hyperbolic equation [26][27][28][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. e rest of this paper is organized as follows. Some basic notations and lemmas are given in Section 2, 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. e 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.

Preliminary
Let m and n be two positive integers and assume that the space step and time step are h � 1/m and τ � T/n, respec- be the grid function on Ω hτ . We introduce the following difference and averaging operators: e following lemmas (Lemma 1-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.
Let h > 0 and c be two constants. (10) where c is a nonnegative constant.

Lemma 3. Suppose the mesh grids be
where Square both sides of (14) and apply the Cauchy-Schwarz inequality, then which directly implies e first inequality of (15) implies Multiply both sides of (17) by h, and sum for i from 1 to m − 1, then us, we have which implies and ends the proof.

□
Introduce a grid function space on Ω h : en for v ∈ V, define the norms Complexity where Multiplying both sides of (24) by h and summing for j from 1 to m − 1, we get Multiplying both sides of (24) by h/2 and letting j � m in (24), then after combining it with (25), we get Similarly, we have Combining (26) and (27), it follows is completes the proof.

Derivation of the Finite Difference Scheme
Define the grid function Consider equation (1) at the inner points (x i , y j , t k ) in Ω hτ , then it follows According to Lemma 1(e), it follows where R k ij � O(τ 2 + h 2 ). ese 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 at (x m , y j , t k ), it follows Similarly, the boundary condition 1.4 at (x i , y m , t k ) satisfies By (3) and (4), the differential equation at the corner point (x m , y m , t k ) reads According to Lemma 1 (a), (c), and (d), we have (38) Applying (35) and (37) into (32), we get From Lemma 1(f ) we have, Putting (43) into (42), then by Lemma 1(a) and (e), the following equations hold: 2 where r ij � O(τ + h 2 ). At (x m , y j , t 1/2 ), we construct the following difference scheme: where r mj � O(τ 2 + h 2 ). Similarly, at (x i , y m , t 1/2 ) and (x m , y m , t 1/2 ), we have respectively, where r im � O(τ 2 + h 2 ) and r mm � O(τ 2 + h 2 ). For the initial condition and the left and bottom boundaries, we have Finally, by dropping the infinitesimals in (31), (39)-(47), with W k ij replaced by w k ij , we get the finite difference schemes for (1)-(4) as follows (59)

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 that u k ij | 0 ≤ i, j ≤ m, 0 ≤ k ≤ n solves the following difference scheme: then for arbitrary grid ratio λ � τ/h, we have where Multiplying both sides of (60) by 2h 2 D t u k ij , and taking summation for i and j from 1 to m − 1, we get 6 Complexity e first term in (73) implies e second term and the third term in (73) imply respectively. Putting (74), (75) and (76) into (73), we get Considering (61)-(63) and by inequality According to the definition of E k , (77) implies If we confine τ ≤ (2/3), then By Gronwall inequality in Lemma 2, we get Next we estimate E 0 . Multiplying (64) by h 2 δ t u 1/2 ij , and summing up for i and j from 1 to m − 1, we have which similarly implies 2 8 Complexity According to the definition of E 0 , we obtain Putting (85) into (82) and noticing kτ ≤ T, we achieve (70). is completes the proof. □ □

Existence, Convergence, and Stability
In this section, we will discuss the unique solvability, convergence, and stability of the difference scheme (50)-(59).
According to the principle of induction, the difference scheme of (50)-(59) is uniquely solvable for all w k (1 ≤ k ≤ n). is completes the proof. □ Define the pointwise error by e following theorem states the convergence of difference scheme (50)-(59).

Theorem 3.
Suppose that w(x, y, t) ∈ C 4,4,3 x,y,t (Ω × [0, T] is the solution of (1)- (4) and w k ij | 0 ≤ i, j ≤ m, 0 ≤ k ≤ n is the solution of difference scheme (50)-(59). en there exists a constant C T such that where Taking into account the order of r ij and R k ij (0 ≤ i, j ≤ m, 1 ≤ k ≤ n), and the fact that e 0 ij � 0, there exists a generic constant c such that       Tables 1-4. theoretical analysis of eorem 3. When the temporal size is fixed with τ � 1/200, Table 2 gives the discrete L 2 -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 eorem 3. e convergence results with fixed grid ratio λ � 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 eorem 4. e curve of convergence order in Tables 1-4 is shown in Figure 2.

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 discrete L 2 -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. e theoretical analysis method can be applied to three-dimensional problems accordingly.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.