Study On Structure-Preserving Algorithms Of The Klein-Gordon-Zakharov Equation

With the rapid development of science and technology,more and more physical phenomena can be described by nonlinear development equations.In many cases these systems are conserved,so the construction of some numerical algorithms that can preserve the system’s conserved quantities has become a hot research topic.In this paper,the algorithm of preserving structure of Klein-Gordon-Zakharov(KGZ)equation is studied.Roughly divided into the following two jobs:Firstly,a series of local preserving structure algorithms are constructed systematically for one-dimensional classical KGZ equation by using complex construction method,including four polysymplectic algorithms,four local energy conservation algorithms and four local momentum conservation algorithms.For one-dimensional classical KGZ equation,local energy conservation algorithm and local momentum conservation algorithm have not been studied before,and the existing Dosin algorithm is also discussed separately.The local preserving structure algorithm constructed in this paper treats the time direction and space direction equally,and extends the symplectic structure of the whole time layer to the local region,even to every point.It overcomes the dependence of global structure-preserving algorithms on boundary conditions and is a generalization of global structure-preserving algorithms.In particular,under appropriate boundary conditions,local structure-preserving algorithm is naturally global structurepreserving algorithm,but vice versa.The stability analysis of the proposed scheme is also carried out.It is proved that the explicit scheme is conditional and the implicit scheme is unconditional.At the end of the paper,our numerical experiment also verifies the theoretical analysis.Second,we construct a meshless structure-preserving algorithm for the d-dimensional(d=1or 2)Special KGZ equation.Based on the method of line,we first use the radial basis function method of the quasi-interpolation to discrete the space direction,which can be cast into Hamiltonian via using the premultiplication of a diagonal matrix.Then,the average vector field method is used to discretize the time direction and a fully discrete structure-preserving scheme is finally obtained.The energy preservation property of the fully discrete meshless preserving structure algorithm is proved in detail.Numerical experiments on uniform,non-uniform and random grids validate the above theoretical analysis.