The Error Estimates Of LDG Method Based On Generalized Numerical Fluxes For Convection Diffusion Equations

As a special kind of finite element method,local discontinuous Galerkin(LDG)method is widely applied to solve numerical solution problems of high-order partial differential equations due to its advantages of strong stability and high accuracy.It is extremely important to select appropriate numerical fluxes for LDG method to ensure the low numerical viscosity of numerical scheme in the course of the computation.This paper mainly studies the error estimates of the LDG method for convection diffusion equations based on generalized numerical fluxes.Firstly,the paper focuses on the stability analysis and error estimates of LDG method for nonlinear convection diffusion equations.When generalized local Lax-Friedrichs numerical fluxes(GLLF)are used for convection term and generalized alternating numerical fluxes are selected for diffusion term,the L~2stability of the LDG scheme of the equations is acquired from the monotonicity of the numerical fluxes of the convection term.For the sake of further dealing with nonlinear term in the process of error analysis,linearization technique is employed.By designing and analyzing a special kind of global Gauss-Radau(GGR)projection,the errors that arise from boundary condition are eliminated.Combined with a priori assumptions,it is proved that the LDG scheme of the equations reaches k(10)1 order of convergence.Secondly,we mainly pay attention to the error estimates of LDG method for variable coefficient convection diffusion equations where the diffusion term coefficient is a nonlinear function.Due to the particularity of numerical fluxes selected by the LDG scheme of variable coefficient equations,the GGR projection with correction term is established in the process of error analysis to eliminate the errors generated by the linearized variable coefficient term at the boundary.Combined with a priori assumptions,we acquire the k(10)1 order convergence accuracy of the LDG scheme.Finally,according to the theoretical analysis,some suitable numerical examples are selected to carry out numerical experiments,and the experimental results show that the LDG scheme under generalized numerical fluxes can achieve k(10)1 order convergence accuracy for nonlinear and variable coefficient convection diffusion problems.In addition,experimental data show that the error estimates are still valid for non-monotone numerical fluxes of convective term.