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 discretized norm. The theoretical result is verified by a numerical experiment.
1. Introduction
In this paper, we consider the finite difference discretization for the following initial boundary value problem (IBVP) of wave equation in a square :where , , and are given sufficient smooth functions, and , , , 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 solution represents the transverse displacement of the membrane, and in models for acoustic pressure fields, the solution represents the fluid pressure (see, [1, 2] for more examples). The boundary conditions on the right and top sides of 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 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 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. 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
Let and be two positive integers and assume that the space step and time step are and , respectively. Define , and . Let be the grid function on . We introduce the following difference and averaging operators:
The 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 and be two constants.(a)If , then(b)If , then(c)If , then(d)If , then(e)If , then(f)If , then
Lemma 2. (Gronwall Inequality). Suppose are nonnegative sequences such that , thenwhere is a nonnegative constant.
Lemma 3. Suppose the mesh grids be , where , . Let be mesh function on such that , thenwhere , , .
Proof. For , we haveSquare both sides of (14) and apply the Cauchy–Schwarz inequality, thenwhich directly impliesThe first inequality of (15) impliesMultiply both sides of (17) by , and sum for from 1 to , thenThus, we havewhich impliesand ends the proof.
Introduce a grid function space on :
Then for , define the norms
Lemma 4. Let be mesh function on such that for , then we havewhere .
Proof. From Lemma 3, we haveMultiplying both sides of (24) by and summing for from 1 to , we getMultiplying both sides of (24) by and letting in (24), then after combining it with (25), we getSimilarly, we haveCombining (26) and (27), it followsThis completes the proof.
3. Derivation of the Finite Difference Scheme
Define the grid function by
Consider equation (1) at the inner points in , then it follows
According to Lemma 1(e), it followswhere . 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 at , it follows
Similarly, the boundary condition 1.4 at satisfies
By (3) and (4), the differential equation at the corner point reads
According to Lemma 1 (a), (c), and (d), we have
Applying (35) and (37) into (32), we getwhere . Similarly, we havewhere and .
When , equation (1) at initial grids satisfies
From Lemma 1(f) we have,
Putting (43) into (42), then by Lemma 1(a) and (e), the following equations hold:where . At , we construct the following difference scheme:where . Similarly, at and , we haverespectively, where and . For the initial condition and the left and bottom boundaries, we have
Finally, by dropping the infinitesimals in (31), (39)–(47), with replaced by , we get the finite difference schemes for (1)–(4) as follows
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 that solves the following difference scheme:then for arbitrary grid ratio , we havewhere
Proof. DefineMultiplying both sides of (60) by , and taking summation for and from 1 to , we getThe first term in (73) impliesThe second term and the third term in (73) implyrespectively. Putting (74), (75) and (76) into (73), we getwhereConsidering (61)–(63) and by inequality , we getAccording to the definition of , (77) impliesIf we confine , thenBy Gronwall inequality in Lemma 2, we getNext we estimate . Multiplying (64) by , and summing up for and from 1 to , we havewhich similarly impliesAccording to the definition of , we obtainPutting (85) into (82) and noticing , 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 that is known by the initial condition. The difference scheme for is (54)–(59). We consider the following homogeneous equations of (54)–(59):Multiplying (86) by , summing up for and from 1 to , and combining with (87)–(90), we getwhich implies . Thus, is uniquely solvable.
Now suppose that and are uniquely solvable. The difference schemes for satisfy (50)–(53) and (59). The homogeneous equations read:Multiplying (92) by , and summing up for and from 1 to , and applying equations (93)–(96), we havewhich means that . Thus is uniquely solvable.
According to the principle of induction, the difference scheme of (50)–(59) is uniquely solvable for all . This completes the proof.
Define the pointwise error by
The following theorem states the convergence of difference scheme (50)–(59).
Theorem 3. Suppose that is the solution of (1)–(4) and is the solution of difference scheme (50)–(59). Then there exists a constant such thatwhere .
Proof. Subtracting (50)–(59) from (31), (39)–(41) and (44)–(49) respectively, we have the following error equationAccording to Theorem 1, we getwhereTaking into account the order of and , and the fact that , there exists a generic constant such thatThis completes the proof.
According to Lemma 4, we can conclude that , namely, the difference solution is of second-order convergence in norm.
Theorem 4. For any fixed grid ratio , the difference scheme (50)–(59) is stable with respect to the initial value , , and the source term . More precisely, we havewhere .
Proof. Suppose that the solution of difference scheme (50)–(59) is , then for any we can get similarly to Theorem 1whereTaking account into the fact that , , there exists constant such thatwhich 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 is defined bywith initial conditions and designed as
For this specified problem, the exact solution is
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.
(a)
(b)
Let us denote the error by
Table 1 provides the computed results with a fixed . The data demonstrate that the temporal convergence order of difference scheme (50)–(59) is second order in discrete –norm, which is in accord with theoretical analysis of Theorem 3. When the temporal size is fixed with , Table 2 gives the discrete –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.
The convergence results with fixed grid ratio and 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 , 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.
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 discrete –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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the NSF of China (nos. 11901365 and 11772177).