Discontinuous Galerkin Methods For The Two Dimensional Zakharov-Kuznetsov Equation And The Distributed Order Fractional Diffusion Equation

Discontinuous Galerkin(DG)methods are a class of finite-element methods using completely discontinuous basis functions,which are usually chosen as piecewise polynomials.The method can not only maintain the advantages of finite element and finite volume,and overcome the shortcomings.The boundary does not require continuity,allowing the existence of "discontinuity",which makes it very easy to deal with complex boundary and boundary value problem;the allowance of arbitrary triangulation with hanging nodes;it is very suitable for h-p adaptive computing,and has high parallel efficiency.With the development of computer technology,the discontinuous Galerkin(DG)methods have been widely applied in the field of science and computing,and have attracted more and more attention.In this paper,the discontinuous Galerkin methods for the two-dimensional Zakharov-Kuznetsov equation and the distributed order diffusion equation is studied.This paper consists of three sections.In the first section,we introduce the background of discontinuous Galerkin methods,two-dimensional Zakharov-Kuznetsov equation and the distribution order diffusion equation,and also give some definitions about the spaces and inequality which will be used in theoretical analysis.In the second section,we propose and analyze a stable and efficient discontinuous Galerkin method for 2D Zakharov-Kuznetsov equation,and prove the property of the scheme.Numerical experiments show that the scheme is convergent and effective.In the third section,we study the discontinuous Galerkin method for the distributed order diffusion equation,The proposed method is based on a finite difference scheme in time and discontinuous Galerkin methods in space.Theoretical analysis shows that the method is unconditionally stable and convergent.The results of numerical experiments show that our scheme is convergent.The results show that the discontinuous Galerkin method for the two dimensional Zakharov-Kuznetsov equation and the distributed order diffusion equation is stable and efficient,and is a good tool for numerical simulation of such problems.