Study On The Structure-Preserving Algorithm For Two Kinds Partial Differential Equations In Plasma Physics

Partial differential equations are an important class equations in the mathematical world,descrbing some important physical phenomena in the real world.However for most partial differential equations,it is very difficult or even impossible to find the analytical solutions.Therefore,the numerical solution of partial differential equations is very meaningful,On this basis,Mr.Feng Kang proposed the idea of structure-preserving algorithm,that is,the numerical scheme should keep the intrinsic properties of the original equation as much as possible.This paper is based on this idea,numerical studies have been carried out on the Zakharov system and the Lorentz force system respectively.The details are as follows:First,a local structure-preserving algorithm is constructed for the Zakharov system using the finite difference method.This scheme can maintain local momentum conservation and local mass conservation.The local structure-preserving algorithm considers both time and space directions,composite construction,and it is an extension of the global structure-preserving algorithm.Compared with the global structure-preserving algorithm,it’s advantage is that the local structure-preserving algorithm does not depend on specific boundary conditions,but can maintain structure conservation in any time-space region.When the boundary conditions are suitable,the local structure-preserving algorithm can also satisfy the corresponding global conservation law.Secondly,based on the idea of line method,two meshless energy-preserving algorithm is constructed,using the radial basis function method in the space direction to get the semi-discrete scheme,but this semi-discrete scheme is not a hamiltonian system,then we left multiply the both side of the semi-discrete system with a diagonal matrix which make an equivalent system,and the eauivalent system is hamiltonian system.Then using the energy-preserving intergral discrete the time direction to get an energy-preserving scheme.The advantage of this scheme is that the scheme is constructed on the scattered points,that is,this scheme is meshless.This scheme can keep the discrete global energy and global mass of the Zakharov system unchanged.Finally,an energy-preserving scheme is constructed using the invariant energy quadratization method and the symplectic Runge-Kutta method for the Lorentz force system.The advantage of this scheme is that it not only maintains energy conservation,but also the scheme can be arbitary high-order.