Some Properties Of Discontinuous Galerkin Methods For Several Evolution Equations

In this thesis,the main contents are properties of discontinuous Galerkin methods for several evolution equations,including superconvergence for 1D convection-diffusion equations and optimal error estimates for 2D nonlinear hyperbolic equations.Evolution equation is a general term of partial differential equations with time variables and their derivatives,which is often used to describe the state or process evolving with time in practice.With the progress of science and technology,evolution equations have been widely used in biology,fluid mechanics,environmental science and other fields,and the study of their numerical solutions has guiding meanings for people to understand natural phenomena.The choice of numerical fluxes in the discontinuous Galerkin method not only has a direct effect on the stability of the scheme,but also affects some characteristics of the numerical solution.Since the generalized fluxes take the weighted average of numerical solution from both sides at the cell interface,the numerical viscosity of the scheme can be controlled by adjusting the fluxes parameters,which makes it easier to construct discontinuous Galerkin scheme for solving complex systems.From the view of computation,higher order accuracy of numerical solution and its derivative at some points can be obtained by using the superconvergence properties without increasing the amount of computation and storageThe main contents can be summarized as follows:Firstly,the superconvergence of local discontinuous Galerkin methods based on generalized alternating numerical fluxes is analysed for 1D linear convection-diffusion equations.The core idea of this work is to construct a special interpolation function and establish supercloseness between numerical solution and the interpolation function,then the cell averages,numerical fluxes in the discrete L2 norm and the function value as well as its derivative at generalized Radau points are shown.The technique of constructing the correction functions is to use the switch of the time derivative and spatial derivative through the integral operator in combination with integration by parts,and the special interpolation function can be obtained by modifying the generalized Gauss-Radau projection.By choosing suitable numerical fluxes,the superconvergence results are valid for the mixed boundary conditions and Dirichlet boundary conditions.Secondly,the local discontinuous Galerkin method with Godunov flux and alternating numerical fluxes is proposed to study the superconvergence for 1D nonlinear convectiondiffusion equations.In order to completely eliminate the influence of the integral terms and boundary terms on superconvergence when the direction of flow changes,a reference flux related to the exact solution is introduced,and a new projection is constructed to deal with the projection errors coming from auxiliary variable and nonlinear convection term.By means of the properties of the new projection and the initial discretization,the supercloseness between the numerical solution and the designed interpolation function is obtained,then superconvergence results of numerical fluxes in the discrete L2 norm and cell averages are proved.By modifying the new projection,the theoretical results are extended to generalized alternating numerical fluxes and more complex boundary conditions.Finally,for two-dimensional nonlinear scalar conservation laws,stability as well as optimal error estimates of DG methods based on generalized local Lax-Friedrichs fluxes are investigated.A special piecewise global projection pertaining to wind variation of physical flux functions is constructed,which breaks the global coupling property along some element edges,and a sharp bound for projection error terms is obtained by combining the special structure of Cartesian meshes.In the error analysis,Taylor expansion linearization technique and reasonable priori assumption are used to deal with nonlinear term.The idea of constructing a special projection can be extended to DG methods with upwind-biased fluxes for 2D linear variable coefficients hyperbolic equations.Numerical experiments for above issues are provided,and the results of the numerical experiments are compatible with the conclusions of the theoretical analysis,which show that the conclusions of theoretical analysis are valid.