Application Of Discontinuous Galerkin Method In Solving Some Differential Equations

Numerical simulation is a branch of physics research,which uses numerical algorithms to solve physical problems for verification and design aid.For complex problems,analytical methods often fail to provide analytical solutions,and experiments consume a lot of resources.Therefore,numerical simulation has been widely developed because of its efficiency and convenience.Discontinuous Galerkin method(DG)is a kind of numerical method.For differential method problems to be solved,the weak solution form is simplified into a system of linear equations by means of test function expansion and flux continuity condition on the boundary.Because of its easy solution and high applicability,it has been widely used in the field of numerical algorithms,and is one of the popular algorithms for solving differential equations.In this paper,the realization of solving several differential equations by DG is studied,including Maxwell equation,Poisson equation and drift diffusion equation.Including,the implementation of hexahedral mesh DGTD(Discontinuous Galerkin Time-Domain)algorithm,realization of UPML(Uniaxial anisotropic Perfectly Matched Layer)based on hexahedral mesh in DGTD algorithm,realization of mixed grid(tetrahedron and hexahedron)DG algorithm.DG solution to the electromagnetic in the source region of ground nuclear explosion,free carrier drift diffusion model.In this paper,the numbering methods of hexahedral mesh are analyzed,the numbering methods of nodes,edges and faces are given,and a group of edge basis functions are constructed.The analytical formula of hexahedral element matrix is given,and the DGTD algorithm of hexahedral mesh is realized.The implementation of hexahedral mesh UPML in DGTD is studied,and its time-domain iteration formula is derived.The results show that it has good absorption effect on outward traveling wave.For three dimensional electromagnetic problems,tetrahedral mesh is often used in DGTD.If the electromagnetic problem includes multi-scale models,the number of grids is large,which reduces the computational efficiency.Therefore,in this paper,based on the study of hexahedral DGTD,combined with tetrahedral DGTD,a virtual element hybrid grid DGTD algorithm is presented.In the algorithm,the model is divided into two subdomains,which are divided by tetrahedral mesh and hexahedral mesh.The tetrahedron is used for the fine structure region near the model,and the hexahedron is used for the peripheral region such as the absorption boundary.Virtual elements are set at the subdomain interface to exchange flux,thus reducing the computation time and unknowns.The errors of numerical results and commercial software are discussed,and the number of mesh and calculation time are reduced compared with tetrahedral DGTD algorithm.The simulation of electromagnetic environment of ground nuclear explosion source area is a problem of multiple physical fields.In this paper,the Compton current model is used to combine Maxwell equation and air ionization equation.Charged particles generate current density in the process of motion,which is the source of current in Maxwell equation.The density of charged particles also changes the conductivity in the air.The calculation area is divided into two sub-areas above the ground and below the ground.The time domain iterative equation of discrete above-ground and underground area by DG method are given.The errors of calculation results and reference results are discussed,and the effects of different distances and depths on the amplitude of electric field are analyzed.Drift diffusion model is often used to describe the transport process of free carriers in semiconductors.This paper presents the solving process of DG algorithm.The time domain iteration formulas of Poisson equation,electron density drift diffusion equation and hole density drift diffusion equation are derived.In the process of carrier movement,the density will be reduced by collision,and the carrier recombination rate is a nonlinear term that changes with the density.In this paper,Newton iteration method is used to solve the recombination rate.The explicit time step iteration of DG is modified to semi-explicit and semi-implicit iteration.Calculate the current density-voltage curve of solar cell and discuss its error with the reference result.