Analysis Of The Implicit-Explicit Ultra-weak Discontinuous Galerkin Method For Convection-diffusion Problems

The ultra-weak discontinuous Galerkin(UWDG)method is a new discontinuous Galerkin(DG)method for solving convection-diffusion equations and time-dependent partial differential equations with high-order spatial derivatives.Most of the current research work focuses on the theoretical analysis of the semi-discrete UWDG method,while the theoretical analysis of fully discrete UWDG method is relatively lacking.In order to solve this problem,this paper will mainly carry out the theoretical research of the fully discrete UWDG method.The implicit-explicit(IMEX)time discretization method is an ideal time discretization method,which can not only avoid the strict time step limitation of the pure explicit time discretization method,but also overcome the problem of slow efficiency of the pure implicit time discretization method.In this paper,the stability,error estimates and superconvergence properties of the implicit-explicit fully discrete UWDG method(IEMX-UWDG method for short)will be studied by using the one-dimensional linear convection-diffusion equation with periodic boundary conditions as the model.The main research contents are as follows:Firstly,the optimal error estimates of the semi-discrete UWDG method for special numerical fluxes are given,combined with a Runge-Kutta type IMEX time discretization method that explicitly deal with convection terms and implicitly deal with diffusion terms,the almost unconditional stability of the corresponding fully discrete IMEX-UWDG scheme is analyzed,that is,when the time step(is independent of the mesh size),the numerical solution of the fully discrete scheme satisfies the strong stability of the norm.Meanwhile,taking the third order IMEX-UWDG scheme as an example,the optimal norm error estimate of the fully discrete scheme is analyzed.Secondly,the above theoretical results are extended to the IMEXUWDG method for general numerical fluxes,and the difficulty of error estimation is overcome by defining special projections.Finally,with the help of the correction function technique,a theoretical analysis of the superconvergence properties of the semi-discrete UWDG scheme for special numerical fluxes is given,and it is proved that or-th order superconvergence rate for cell averages and numerical flux of the function,as well as the UWDG solution is superconvergent with a rate of or to the special projection.At a class of special points,the function values and the first and second order derivatives of the UWDG solution are superconvergent with order,,or,,,respectively,where is a polynomial degree.It is also numerically verified that the fully discrete IMEX-UWDG scheme has the same superconvergence property.