A Compact Difference Method for Viscoelastic Plate Vibration Equation

In this paper, a fourth-order viscoelastic plate vibration equation is transformed into a set of two second-order differential equations by introducing an intermediate variable. A three-layer compact difference scheme for the in-itial-boundary value problem of the viscoelastic plate vibration equation is established. Then the stability and convergence of the difference scheme are analyzed by the energy method, and the convergence order is Finally, some numerical examples are given of which results verify the accuracy and validity of the scheme.


Introduction
Consider the following initial-boundary value problem of the viscoelastic plate vibration equation  viscoelastic media. In [3], the plate vibration equation is derived in detail. In [4], Wang proves  have not seen any articles that solve the initial-boundary value problem of viscoelastic plate vibration equation by compact difference method until now. The goal of this paper is to construct a compact implicit difference scheme for the problem (1.1). The outline of this paper is as follows. In Section 2, the fourth-order viscoelastic plate vibration equation is transformed into a second-order system of equations. The space derivative terms of the equations are discretized by the fourthorder compact difference scheme, while the time derivative terms are discretized by the second-order central difference scheme. Finally, the compact implicit difference scheme of (1.1) is constructed. In Section 3, we prove that the presented compact difference scheme is convergent and unconditionally stable by the energy method. In Section 4, some numerical examples are given to verify the accuracy of the scheme, which show that the scheme has high practicability.

Compact Difference Scheme
In this section, we develop a compact difference scheme of the problem (1.1), , , , , , 0, , From the Taylor expansion, we obtain , , , , , 12 Substituting the above four formulas into (2.2) and replacing n ij u with ( ) , , , , , Performing the compact operator x A , y A on both sides of (2.3), we get  , , .
Omitting the small terms ( ) .

Stability and Convergence Analysis
In this part, we prove the stability and convergence of difference schemes (2.15) by energy method. We introduce the space Engineering x In order to give stability and convergence analysis of the difference scheme (2.15), the following lemmas are given.

Lemma 3.1 [13] For any grid function
Lemma 3.2 [13] [17] For any grid function For convenience, we set  There is the stability estimate Engineering Taking the inner product on both sides of (3.5) with Multiplying the both sides of (3.4) by a and multiplying the both sides of (3.6) by 1 2 , then summing the results and applying Lemma 3.1, we obtain ( )  Noticing 0 µ > , 0 a > , summing over n from 1 to m on both sides of (3.8) and applying Cauchy-Schwarz inequality, we get   . Then, we estimate the right side of (3.12).
Noticing the initial conditions, we know 0 0 The theorem has been proved.

Numerical Experiment
In this section, we present two numerical examples. In addition to showing convergence orders and the effectiveness of the presented difference scheme (2.15), we apply the difference scheme to a practical problem and the results show the effect of the viscosity coefficient on the plate vibration. Example 1. In this example, let 2 1 t < ≤ , the analytic solution is chosen to be ( ) , , cos sin sin u x y t t x y π π = π , then ( ) 2 , , 2 cos sin sin v x y t t x y = π π π π , and the source term is ( ) , , 2 sin cos 4 cos 2 cos sin sin . f x y t t t t t x y π π π π π = − − + π π π + π Taking the space step 1 1 x y h M M = = , and the time step 1 N τ = , we solve the problem by using the presented compact implicit difference scheme (2.5). Table  1 shows the error and spatial convergence order of the numerical solution U. Table 2 shows the error and time convergence order of the numerical solution U. Figure 1 and Figure 2 show the images of the numerical solution U and the analytic solution u when the mesh is divided into 32 It can be seen from the tables that in the sense of the maximum norm and the L2 norm, the time convergence order has reached the second order, and the space convergence order has reached the fourth order, which shows the validity of the compact difference scheme established.
Given below is the images of the numerical solution and the analytic solution Example 2. In this example, let  . We consider the problem which is more realistic in sense.
We observe vibration at the center point of the plate when the viscosity coefficient µ takes different values. Setting 1 k = , Figure 3 and Figure 4 show the images of displacement u when µ is set as 1 and 10 respectively. It can be seen from the figures that using the compact difference method to solve the problem (1.1) well shows the process of damped vibration. And the figures depict the effect of damping caused by the viscous term on vibration. We can see the values of   the amplitude of the center point of the plate in different time periods. The amplitude becomes smaller and smaller as time goes on, and speed of decreasing amplitude becomes faster and faster with increase of the viscosity coefficient.

Conclusion
In this article, we develop a compact difference scheme for the viscoelastic plate vibration equation. The fourth-order differential equation is transformed into a second-order differential equation system by introducing intermediate variables, then the spatial derivative and time derivative are discretized by fourth-order compact difference method and the central difference method respectively. The stability and convergence of the scheme are proved by using the energy method. The difference scheme is stable and convergence rate is ( ) 2 4 O h τ + . The numerical results verify the validity and accuracy of the scheme. In future work, we will consider the viscoelastic plate vibration equation with fractional derivative.

Funding
The work is supported by Shandong Provincial Natural Science Foundation, China (ZR202102250267).