Optimal Error Estimates Of Discontinuous Galerkin Methods For Several Types Of Equations

In this dissertation,the main task is the optimal error estimates of the discontinuous Galerkin method for partial differential equations.The equations studied are hyperbolic conservation laws and linearized Korteweg-de Vries equations,which are widely applied in many research fields.The discontinuous Galerkin methods are a class of numerical methods for partial differential equations,whose solution have arbitrary high order of accuracy when solving the smooth solution problem,and can precisely capture the dis-continuity of exact solution when solving the discontinuous problem.The discontinuous Galerkin methods have been widely used in many research fields due to the excellent properties.Numerical flux is a key ingredient of discontinuous Galerkin methods,and the fluxes we study in this dissertation are the generalized fluxes.Compared with the upwind flux,the generalized fluxes provide more flexibility to the numerical schemes,and make the schemes to hold adjustable numerical viscosity so that the schemes are suitable in solving different equations.As the generalized flux contains information of numerical solution from both sides at the cell interface,so the analysis to the convergence of the scheme with generalized flux is more complex,especially on the projection that is used in the error estimates.By constructing proper projection with analysing its properties,we obtain the optimal error estimates of the discontinuous Galerkin method for the equations.The main contents can be summarized as follows:Firstly,the discontinuous Galerkin method with upwind-biased flux(which is the generalized flux)for linear hyperbolic equation with variable coefficient is presented,and the optimal error estimate property of the method is analysed.As the wind direction of the equation is not fixed,the global projection is not appropriate in the error estimates due to its existence may not be true.In this dissertation,the piecewise global projection is proposed based on the characteristic of the equation with variable coefficient.The projection is decoupled at the special cell interface,hence the projection holds some properties and is suitable in the analysis for this problem.By virtue of the projection,the optimal error estimate property of the discontinuous Galerkin method can finally be analysed.Secondly,the discontinuous Galerkin method with generalized local Lax-Friedrichs flux for scalar nonlinear hyperbolic conservation law equation is presented,and the opti-mal error estimate property of the method is analysed.In the error estimate for nonlinear equation,large quantity of linearization technique is applied.As the generalized local Lax-Friedrichs flux contains information of numerical solution from both sides at the cell interface,the terms in error equation after the linearization is more complex.In addition,the proper priori assumption is necessary in the estimate to the terms of error equation,then in combination with the piecewise global projection,the optimal error estimate property of the discontinuous Galerkin method can be analysed.Thirdly,the local discontinuous Galerkin method with generalized fluxes for lin-earized Korteweg-de Vries equation is presented,and the optimal error estimate property of the method is analysed.The fluxes for different parts of the scheme are independent,which provides more flexibility to the scheme but also increases the difficulty to the sta-bility analysis.Moreover,a new numerical initial condition is constructed,which fits for the scheme as well as holds the optimal error estimate at the initial time.Combine with the conclusion of stability analysis,the numerical initial condition and the global projection,the optimal error estimate of the local discontinuous Galerkin method can be analysed.Large amount of numerical experiments are provided in this dissertation.The results of the numerical experiments is compatible with the conclusions of the theoretical analysis,which confirm the validity of the theoretical conclusions.