Optimal Point-wise Error Estimates Of Finite Difference Methods For Solving The Two-dimensional Sine-gordon Equation

The sine-Gordon(SG)equation has rich geometric and physical background and is an important class of nonlinear dispersion equations.Therefore,it is very meaningful to solve the equation numerically.However,the existing algorithms are developed for the one-dimensional case.Since the original embedding inequality is no longer valid in the two-dimensional case,it is difficult to obtain the optimal error estimate of the algorithm under the infinite norm,and related results were not found in the literature.In this paper,numerical study of the two-dimensional SG equation is carried out,three finite difference schemes are constructed,and the optimal error estimation and convergence proof are given.First,this paper extends the four-layer explicit conservation difference scheme proposed in the existing literature for the one-dimensional SG equation to the two-dimensional case,and proves that the scheme maintains the energy conservation property of the original problem in the discrete sense,and at the same time,the restriction on grid ratio in the existing convergence results is relaxed,and the optimal error estimation of the format is established in the sense of maximum norms.Then,we propose an alternating direction implicit(ADI)difference scheme for the two-dimensional SG equation.Without any requirement for the grid ratio,we establish the optimal error estimate in the sense of maximum norm.Finally,a new nonlinear conserved difference scheme is proposed for the two-dimensional SG equation.Its unique solvability and the conservation property in the discrete sense are proved,and the optimal error estimation in the maximum norm sense is established without any requirement for the grid ratio.The point-wise convergence orders of the three schemes are all second order in the space and time directions.The key techniques used in the convergence analysis,except for the en-ergy method,including mathematical induction,etc.And numerical examples verify the error estimation and conservation properties.