Energy-preserving Numerical Methods With Optimal Error Estimates For The Klein-Gordon Equation In High Dimensions

The Klein-Gordon(KG)equation,as one of the most basic equations used to describe particles with zero spin in relativity quantum mechanics and quantum field theory,has an irreplaceable practical background and physical meaning.This paper uses the finite difference method and finite element method to numerically study the high-dimensional KG equation and establishes an optimal error estimate for the numerical format.Firstly,we design and analyze a finite difference scheme to solve the nonlinear Klein-Gordon(KG)equation in d(d=2,3)dimensions.It is proved that the proposed scheme preserves well the total energy in the discrete sense,which is consistent with the conservative property possessed by the original KG system.Besides the standard energy method,the key techniques used in convergence analysis are the lifting technique and the cut-off function technique.Without any restriction on the grid ratio,we establish the optimal error estimate of the proposed energy-preserving scheme.However,certain restriction on the grid ratio is usually required by those previous works in the literature.The convergence rate is proved to be of O(h2+?2)with spatial grid size h and time step ?.Numerical results are carried out to confirm our theoretical analysis.Secondly,we design and analyze a Galerkin finite element method(FEM)to solve the nonlinear KG equation in d(d=2,3)dimensions.By virtue of Li-Sun's error-splitting technique,we construct and analyze a corresponding time-discrete system,then establish the optimal error estimate of the fully discrete Galerkin FEM without any restriction on the grid ratio,and the convergence rate is proved to be of O(hr+1+?2).Numerical examples show that our theoretical analysis is feasible.