Artificial Boundary Method For Nonlinear Wave Equation

In this paper,we study the artificial boundary method of coupled nonlinear Klein-Gordon equations and sine-Gordon equations on unbounded domains.These two equations are widely used in many important physical fields such as solid state physics,plasma physics and nonlinear optics.The unbounded nature of the physical domain and the nonlinearity of the coupled equations make the existing numerical methods cannot be directly used to solve the corresponding partial differential equations.The artificial boundary method is one of the effective methods to solve the unboundedness of the physical region.Its main idea is to introduce the artificial boundary on the unbounded region,divide the unbounded region into the bounded calculation region and the unbounded outer region,design reasonable and effective artificial boundary conditions on the artificial boundary,and transform the original problem on the unbounded region into the initial boundary value problem on the bounded calculation region.Based on the idea of operator splitting method and overcoming the difficulties brought by nonlinearity in the design of artificial boundary conditions,the original problem is divided into linear and nonlinear subproblems.With the help of the artificial boundary conditions of linear subproblems and the idea of operator splitting,the artificial boundary conditions of coupled nonlinear wave equations are obtained.Thus,the original problem defined on the unbounded domain is simplified to the initial boundary value problem on the bounded computational domain.In order to analyze the stability of the simplified initial-boundary value problem,the corresponding energy functional is defined,and the corresponding stability is analyzed by using the energy conservation property of the coupled nonlinear wave equations.The finite difference method is used to discretize the simplified initial boundary value problem.Finally,the feasibility and effectiveness of the artificial boundary conditions are verified by numerical examples.