Compact Finite Volume Methods For Two Types Of Klein-Gordon Equations

Klein-Gordon(KG)equation is also called Klein-Gordon-Fock equation.As a relativistic form of schrodinger equation,it plays an extremely important role in mathematical physics,especially in nonlinear dynamics,including general relativity,radiation theory,scattering stability and so on.Its numerical solution is one of the hot topics in the numerical solution of differential equations.In this paper,a compact finite volume method is proposed for two kinds of Klein-Gordon equations.This method is derived from the finite difference method.It is based on the idea of compact difference method,and then discretizes the equa-tion by the finite volume method.Finally,a high precision finite volume scheme is obtained.Because of its flexible mesh generation,fewer nodes and high accuracy,it has attracted more and more attention.This paper is divided into four chapters.The first chapter is the introduc-tion,it introduces the physical background of the Klein-Gordon equation and the development of the compact finite volume method,and briefly describes the main structural ideas and structural fralework of this paper.In chapter two,a com-pact finite volume scheme is proposed for one-dimensional Klein-Gordon equation.The linear algebraic equations formed by this format have tridiagonal property and easy to solve.Finally,it is proved that the scheme has fourth order accuracy in spatial direction and second order accuracy in temporal direction according to dis-crete L2 norm and L? norm.Numerical examples also verify the correctness of the theoretical analysis and the validity of the format.In chapter three,according to the same idea as chapter two,a compact finite volume scheme is proposed for one-dimensional linear schrodinger equation.Similarly,the equations formed by this scheme have tridiagonal property.The last numerical example also shows that the discrete L2norm and L? norm have fourth-order accuracy in spatial direction and second-order accuracy in temporal direction.Chapter two and chapter three show that the compact finite volume scheme proposed in this paper has high accuracy and good computational stability.Chapter four is a summary of the whole paper and a prospect of the future work.