Discontinuous Finite Element Method, The Variational Iteration Method And Lagrange Coordinates

Numerical methods for partial differential equations have been a important tools in scientific research areas, such as nature science, engineering techniques and economic administration etc. The researches on the approximate analytical methods and the nu-merical methods have important theory and practical values.Variational iteration method is one of the efficient methods for constructing the approximate analysis solutions of partial differential equations. It is based on the La-grange multiplier. By introducing the restricted variation, the Lagrange multiplier can be easily identified. The advantages of the variational iteration method include inde-pendent little parameter, easy programming and ease to do the analytic computation on the solutions obtained by this method.Discontinuous Galerkin finite element method has sevaral advantages beside those that the ordinary finite element method has, such as local conservation, allowance of discontinuities, formal high order accuracy, easy realization of parallel and adaptive calculation. The local discontinuous Galerkin finite element method is an extension of the Runge-Kutta discontinuous Galerkin method. It has all the advantages of the discontinuous Galerkin method. Local discontinuous Galerkin method has its own ad-vantages too. It can be used to calculate the numerical solutions of some high order partial differential equations. It can get the same order of accuracy for both the nu-merical solution and the variables introduced in calculation.The content of the article includes the following three parts:The first part con-structs the variational iteration method for general Hirota-Satsuma coupled KdV equa-tions and coupled MKdV equations and some useful results are obtained; The second part constructs the local discontinuous Galerkin method based on the Hopf-Cole trans-formation for one and two dimensional Burgers equation; The third part discusses the discontinuous Galerkin finite element method for one and two dimension gas dynamic equations on Lagrangian coordinate.The article is arranged as follows:In chapter 1, we briefly give the background of our research and introduce the development and history of the variational iteration method and the discontinuous Galerkin finite element method. We also introduce the development of the Lagrangian method.In chapter 2, variational iteration method is used to construct the approximate analysis solutions of general Hirota-Satsuma coupled KdV equations and coupled MKdV equations. The numerical examples are used to compare the numerical solutions with the exact solutions. In chapter 3, we develop the local discontinuous Galerkin method based on the Hopf-Cole transformation. One dimension Burgers equation can be transformed to the linear diffusion equation by Hopf-Cole transformation. For a kind of system of two dimension Burgers equations satisfing the potential symmetry conditions, it can be transformed to two dimension diffusion equation by Hopf-Cole transformation. The two dimension Burgers equation can be rewrite as a system of two dimensional Burgers equations satisfing the potential symmetry conditions. The local discontinuous Galerkin method are used to find the numerical solution of the diffusion equation. Then solutions of the original Burgers equation(s) can be obtained by the Hopf-Cole transformation. Using the LDG method we can calculate the numerical solutions of unknown function and the auxiliary variable simultaneously. There is no need to reconstruct the deriva-tive used in the Hopf-Cole transformations. The accuracy of the numerical solutions for original and auxiliary variable calculated by LDG method is same, which can avoid introducing the extra errors.In chapter 4, we discuss the discontinuous Galerkin methods for compressible gas dynamic equations. We begin with the conservational form of the gas dynamic equa-tions in Lagrangian coordinate. We extend the Runge-Kutta discontinuous Galerkin finite element method in Eulerian coordinate to the Lagrangian coordinate. Some sin-gle medium and multi medium fluid examples are used to demonstrate the efficiency and practical usefulness of our scheme. The method solves the geometry and physical conservation laws simultaneously. There,is no need to use the grid velocity and the staggered meshes. The method is simple and easy to use. It combines the advantages both Lagrangian method and the discontinuous Galerkin method, which are high order accuracy and good numerical simulation results for contact discontinuity.