The Finite Difference Schemes For Two-dimensional Wave Equations With Two Kinds Of Boundary Conditions

The wave equation with different boundary conditions is important research objects of distributed parameter control theory,and widely used in various fields such as power,chem?ical,materials,information,biology,and economics.However,due to the complexity of boundary conditions of the wave equation,analytical solutions are often not easy to find,so the study of numerical solution about wave equations have theoretical and practical re-search significances.In this paper,the finite difference method are used to study difference schemes about two-dimensional wave with two kinds of boundary conditions.In the first part,it mainly considers the following the initial boundary value problem for the two-dimensional wave equation with the left boundary being zero,and the right boundary being the Robin type boundary condition.Firstly,the two-dimensional wave equation is discretized in the time and space direction,and a three-layer second-order implicit difference scheme can be obtained.Secondly,a priori estimate inequality is constructed by using discrete multiplier method,and then solvability,stability,and convergence in the sense of infinite dimensional norms about the numerical solution of the implicit scheme is proved.At the same time,it can be proved that the accuracy of the scheme in the time and space directions is two order.Finally,a numerical example is used to verify the theoretical results.In the second part,it mainly deals with the initial boundary value problem of the two-dimensional wave equation with Dirichlet boundary value conditions as follows.A new class of weighted average finite difference schemes is constructed.Firstly,a second-order explicit difference scheme for the equation is established.Then a new difference quotient scheme is given to the second-order partial derivative of the space,and a new im-plicit difference scheme is established for the equation.The weighted average of the explicit and implicit difference scheme can obtain a weighted difference scheme,and the scheme is explicit.Secondly,the Fourier method is used to prove that the weighted scheme is stable under the condition of step size ratio,and the accuracy in both time and space directions is third-order.Finally,a numerical example is used to verify the theoretical results.In the third part,a high order is still established for the two-dimensional Robin-type damped boundary wave equation in the first part.The second-order convergence of the scheme in time and the fourth-order convergence in space are verified by Matlab.