The Study Of Exponential Wave Integrator Fourier Pseudo-spectral Methods For The Klein-Gordon Equation

Klein-Gordon(KG)equation is one of the basic mathematical models describing relativistic quantum mechanics.It plays an important role in physics and mathematics,and the study of its numerical solution is also a subject of widespread concern in the field of computational mathematics.It is found that the predecessors who used the exponential wave integrator Fourier pseudo-spectral methods to solve KG equation,mainly focused on second-order numerical scheme.However,high-precision algorithms are sometimes required in practical problems.Therefore,I will propose a class of high order exponential wave integrator Fourier pseudo-spectral methods.The numerical methods use the Fourier pseudo-spectral methods to discretize the equation in space and use the exponential wave integrator methods in time.The important characteristics of these methods are that the step size is not restricted by Courant-Friedrichs-Lewy(CFL)condition,and at the same time the methods have high order in time.Firstly,this paper introduces some basic definitions and lemmas.Secondly,this paper improves the previous methods of constructing the second-order numerical scheme and proposes using an exponential wave integrator Fourier pseudo-spectral method to construct the fourth-order numerical scheme,which is completely explicit and symmetric.In error analysis,the convergence of the numerical method is proved in detail,with fourthorder accuracy in time and spectral accuracy in space,and the step size is not restricted by CFL condition.Then,this paper proposes a higher order numerical scheme,which has 2k(k>2)order numerical scheme,which is also completely explicit and symmetric.In error analysis,the numerical method is proved to have 2k(k>2)order accuracy and spectral accuracy in time and space,respectively,and the step size is not restricted by the CFL condition.Finally,the correctness of the theoretical results is verified by numerical experiments.