Research Article | Open Access
Jiankang Liu, Suying Zhang, "A New Numerical Approximation Method for Two-Dimensional Wave Equation with Neumann Damped Boundary", Complexity, vol. 2020, Article ID 2020161, 14 pages, 2020. https://doi.org/10.1155/2020/2020161
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 norm. The theoretical result is verified by a numerical experiment.
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  and so on. In , 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 , 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 . As far as hyperbolic conservation law is concerned, there are many distinguished papers focused on this topic, for example, the classical ENO/WENO schemes , 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 , the generalized version of this kind of boundary condition in [3, (2.2b)]) was considered in the context of finite element methods. In , 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 . 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.
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 .
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
According to Lemma 1 (a), (c), and (d), we have
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
4. A Priori Estimate of the Difference Scheme
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 obtain