Compare And Study On Some Numerical Methods Of A Kind Of Semiconductor Model

The discontinuous finite element method is a numerical method which developed from the high-resolution finite difference method and the finite volume method, and has the advantages of them both. It uses completely discontinuous basis functions which are chosen as piecewise polynomials, and the time discretization is achieved by the explicit Runge-Kutta method. Since the end of1980’S, it has attracted the attention of more and more math-ematicians. and has been developed well. In this paper, we develop a local discontinuous Galerkin finite element method for solving a semi-conductor model, and prove the stability. At last, numerical examples are given.This paper consists of five chapters.The first chapter is an introduction. In this part we briefly introduce the origin and the development of the local discontinuous Galerkin finite element method and make a summary of the advantages of this method.In the second part, we apply the local discontinuous Galerkin method to solve the semi-conductor model:The boundary condition is given by Φ, satisfies Dirichlet boundary con-dition: vbias is the Voltage, n, satisfies Periodic boundary conditions, and have the ini-tial condition: 其中,nis the electron concentration,ndis the doping,μ is electron mobility. τ=nμ/e is relaxation parameters,m is the electron effective mass,e is the electronic charge,ε is the dielectric constant,θ=kb/mT,kb is Boltzmann Constant,T is the initial temperature,E is Electric field strength. The semi-discrete scheme is given: Where nh and qh are approximations of n,q∈Vkh.And we make an explanation of the time discretization using the method of Runge-Kutta which Shu raised.This method maintains the stability and high-precision.After that the stability for the time discretization is proved by selecting appropriate numerical flux.In the third chapter,the finite difference scheme:note q=(?)σθnx,to findnh,qh,Φh,and Eh∈Vkh,such that(?)v(x),ω(x),z(x),r(x)∈Vkh,satisfy: are given.The fourth chapter shows numerical examples applying the local discon-tinuous Galerkin finite element method, finite difference method and collo-cation method.And the last part is a conclusion of the paper.