Discontinuous Galerkin Methods With Generalized Fluxes For Several Hyperbolic And Diffusion Equations

With the rapid development of computer technology,the demands for high order accuracy and high computational performance numerical methods become more and more significant for computational mathematicians and computational fluid dynamics mathematicians.The rigorous numerical analysis of several aspects of stability,accuracy and superconvergence properties not only provides a solid theoretical foundation of computational effects of methods,but also produces important evidences of the design and improvement of more general high efficient numerical methods.As a class of high order accuracy methods,the discontinuous Galerkin(DG)method has been widely used in many research fields,due to its ability in capturing shocks,flexibility in dealing with complicated boundary conditions as well as adaptivity with respect to the mesh size.For hyperbolic equations with variable coefficients,nonlinear diffusion equations and nonlinear convection-diffusion equations,stability and optimal error estimates of DG methods with upwind-biased as well as generalized alternating numerical fluxes are investigated.The choice of numerical fluxes plays an important role in the stability,accuracy and superconvergence for the DG methods.Besides,compared to traditional numerical fluxes,generalized fluxes posses adjustable numerical viscosities,which is useful for capturing shock and achieving high order accuracy simulations for smooth solutions.Moreover,for high order wave equations,by choosing downwind-biased fluxes for the convection term with the goal of cancelling the numerical viscosity for the dispersive term,a nearly energy-conserving DG method is thus obtained,which improves the accuracy of long time simulations for waves.The systematic study of DG methods with generalized numerical fluxes in this thesis will broaden the application scope of DG method,and it implies some useful guiding meanings for more engineering problems(such as large eddy simulations).First of all,for two-dimensional hyperbolic equations with variable coefficients,stability as well as optimal error estimates of DG methods based on upwind-biased numerical fluxes are studied.By choosing suitable numerical fluxes,stability of the scheme is guaranteed.Meanwhile,piecewise global projections pertaining to sign variation of physical flux functions are constructed,and optimal approximation error estimates are derived by virtue of different collocation conditions for boundaries.Note that the designed projectioncannot complete eliminate the terms involving projection errors.However,by combining the special structure of Cartesian meshes,a superconvergence result of the projection error is proved.Numerical examples of two-dimensional variable coefficient cases confirm the validity and effectiveness of theoretical results.Secondly,for one-dimensional nonlinear and variable coefficients diffusion equations,the local DG(LDG)methods with generalized alternating numerical fluxes are proposed.To completely eliminate projection error terms,some global projections are constructed and analyzed.By virtue of the projection,optimal error estimates of the exact solution and numerical solution are thus obtained.Finally,for one-dimensional nonlinear convection diffusion equations,the LDG methods based on the local Lax–Friedrichs fluxes and the generalized numerical fluxes are presented.An improved central flux is adopted,which the flux is a generalized numerical flux with simple structure that optimized the LDG scheme.For the numerical solutions of LDG methods with different group of fluxes,the optimal error estimates properties are obtained.The special projection,priori assumption and local linearization technique are the key ingredients in the analysis of error estimates.Numerical experiments for above issues are provided,and the results show that the conclusions of theoretical analysis are valid.