Compact Difference Methods For Two-dimensional Fourth-order Hyperbolic Equation

The two-dimensional fourth-order hyperbolic equations have very important physical background.For example,they can be used to describe the vibration of plates.Hence,the study of numerical methods for this kind of equations has important theoretical and practical significance.In this paper,we mainly study the compact finite difference methods for the initial and boundary value problem of the following two-dimensional fourth-order hyperbolic equation:(?) where Ω=(0,a)×(0,b)(?)R2.In order to construct high accuracy numerical solution formats for the above problem,we rewrite the fourth-order equation as some systems of two second-order equations by introducing new variables and apply the fourth-order compact finite difference methods to discretize the spatial derivatives.Then the compact finite difference schemes is obtained and the error analysis is given.Numerical results are provided to verify the accuracy and efficiency of the schemesIn the second chapter,the fourth-order equation is written as a system of t-wo second-order equations by introducing the second-order spatial derivative as a new variable.In order to discrete every equation of the system,we apply the cen-tral difference method to obtain a second order discretization for the second-order time derivative and the compact finite difference operators to obtain a fourth or-der discretization for the second-order spatial derivatives.Firstly,we construct a three-time-level alternating direction explicit method for the problem and the nu-merical experiment shows that this scheme has satisfying performance.Secondly,considering that the explicit formats are generally conditionally stable,we construct a three-time-level implicit compact difference scheme in the next moment and the convergence analysis is provided.The numerical example proves the effectiveness of the methodIn the third chapter,the fourth-order equation is written as a system of t-wo second-order equations by introducing the first-order time derivative and the second-order spatial derivative as two new variables,respectively.We apply the Crank-Nicolson difference scheme to obtain a second order discretization for the first-order time derivatives and the compact finite difference operators to obtain a fourth order discretization for the second-order spatial derivatives.Thus,we con-struct a two-time-level compact difference scheme,for which the stability analysis and the convergence analysis are provided.The numerical examples prove the per-formance of the method.