Solving A Class Of Nonlinear Wave Equations By LDG Method Based On Generalized Alternating Numerical Flux

Wave propagation is a fundamental form of energy transmission.It appears in many fields of science,engineering and industry.It is of great significance to the earth science,petroleum engineering,telecommunications and defense industries.Due to the complexity of the volatility problem,most of the solution problems cannot be solved accurately,so it is very important to study the numerical method of wave equations.The discontinuous Galerkin(DG)method is a numerical method that combines the features of the finite volume method and the finite element method.The local discontinuous Galerkin(LDG)method is a generalization of the Runge-Kutta discontinuous Galerkin(DG)method.By introducing auxiliary variables,the LDG method can write differential equations with higher derivatives as partial differential equations with only first derivatives.Then we use the DG method for spatial discretization.The LDG method has certain flexibility and advantages,for example,it can be easily designed to any order precision,because the order of precision can be determined locally in each cell grid.It also can be applied to complex mesh regions and h-p adaptive calculations with good parallelism.For the construction of the LDG method,the reasonable use of numerical flux can ensure the stability and high precision of the system.Most of the previous studies used pure alternative numerical flux.The selection of this numerical flux does not have a general rule for the grid.The processing of boundary values is too simple.Y.Cheng,X.Meng,Q.Zhang has analyzed the one-dimensional and two-dimensional linear convection-diffusion equations based on generalized alternating numerical flux studies under the Gauss Radau projection.In this case,the LDG method can make mesh boundaries.The processing of values is more general and can achieve the optimal convergence rate under L~2 norm.Based on the advantages of the LDG method for selecting generalized alternating numerical fluxes,this paper makes an detailed exploration and application of the method in the wave equation,and demonstrates the stability and convergence of the equations(energy conservation)using the LDG method.This paper is divided into five chapters:The first chapter introduces the research background of the wave equation,the main work of this paper and the innovation of this paper;the second chapter mainly introduces the LDG method and the related knowledge used in the research process;In the third chapter,the LDG method based on generalized alternating numerical flux is applied to solve the Burgers equation with Dirichlet boundary conditions.The stability and error estimation are analyzed by numerical analysis and verified by numerical experiments.The fourth chapter solve the one-dimensional second-order wave equation with periodic boundary conditions by the LDG method based on generalized alternating numerical flux.The energy conservation and error estimation are theoretically demonstrated,and the theoretical results are verified by numerical experiments.The fifth chapter establishes and analyzes the LDG method for solving two-dimensional wave equations.By means of triangular mesh and generalized alternating numerical flux,the energy conservation of LDG method is demonstrated by theoretical analysis,and according to the numerical value.The error of the numerical solution shown in the example does not become greater with time.