The Finite Difference Schemes For Two-Dimensional Wave Equation With Damping Term

With the development of control theory,the wave equation with damping term and initial boundary value stabilization control has gradually become an important research content.The research results can be widely used in our daily life.However,the wave equation modeling with the practical problems is often with complex damping,for different damping,the corresponding equation solving methods are not same.And use the simple formula is difficult to find exact solution,numerical analysis provides a possibility of infinite approximation of exact solutions for us.Therefore,the numerical study of two-dimensional wave equation is of great significance in both theory and practical application.In the first part,this paper deals with the initial boundary value problem of twodimensional wave equation with mixed damping boundary,（?）a fully discrete scheme is constructed.For internal nodes,the central difference scheme is used to construct,and for boundary nodes,For qualitative analysis of the scheme,a priori estimate must be given by energy analysis method,and then the existence of the solution is verified.Stability and second-order convergence of space and time directions in the sense of L₂norm.Finally,the theoretical results are verified by an example.The second part,based on the alternating direction method,the initial boundary value problem of the two-dimensional wave equation with Robin type damping boundary,（?）.Alternating direction implicit format（ADI）is constructed.Compared with the full discrete implicit scheme in the first part,the proposed scheme is more concise.The advantage of alternating direction scheme is that the two-dimensional problem can be simplified to one dimension,and then only need to be solved by tridiagonal equations on the time layer.A priori estimator proves that the proposed scheme converges on the first order time 1.5 order of the space in the sense of L₂norm.Finally,the theoretical results are verified by an example.In the third part of the paper,we study the two-dimensional wave equation with internal delay damping from the transformation of boundary damping to internal damping with time delay.（?）.The above equation is established by the central difference scheme for the time-delay term,and the fully discrete three-layer implicit scheme is still established in space.Because it is not possible to find a suitable method for qualitative analysis,the correctness of the difference scheme is verified by a numerical example,and the results show that the difference scheme is correct.It shows the second order convergence under infinite dimensional norm.