Structure-Preserving Discontinuous Galerkin Methods For Some Fluid Dynamics Equations

Mathematical models in the form of partial differential equations are extremely useful tools in mathematical,scientific and engineering communities.Development of robust,efficient and highly accurate numerical algorithms for simulation of their solutions continues to be a challenging task.High order numerical methods,such as discontinuous Galerkin(DG)method and schemes with weighted essentially non-oscillatory(WENO)reconstructions have been under great development for hyperbolic type PDEs with a broad range of applications in the past few decades.One important,yet challenging direction on further development of these high order methods are to ensure the structure preserving properties,i.e.to develop high order numerical methods that preserve certain structures or other fundamental continuum properties of the underlying models exactly.This dissertation consists of three parts.The first part is about the positivity-preserving well-balanced central discontinuous Galerkin-finite element methods of one-dimensional and two-dimensional Green-Naghdi equations.Green Naghdi equations is an approximate model of Euler equations based on shallow water assumption,it is a kind of nonlinear weak dispersion shallow water wave equations.The second part is the positivity-preserving well-balanced central discontinuous Galerkin method for the coupled system of two-dimensional nonlinear shallow water wave equations and sediment transport equations,which is used to study the shallow water flow on erodible river bed.The third part is about the high-order bound preserving discontinuous Galerkin-finite element method for incompressible Navier-Stokes equations with variable density.There are three problems in solving Green-Naghdi equations.One is that the flux and source terms of the model contain mixed derivatives of time and space.The other is that the model,like the nonlinear shallow water wave equations,has a still-water stationary solution.For the stationary solution,the flux of the equations is non-zero,but it is balanced by the source terms.Moreover,the usual numerical methods cannot keep the balance between the flux and the source terms,so when encountering problems related to the stationary solution,numerical oscillations may occur.Third,when the problem involves the dry area or almost dry area,with the movement of water waves,the numerical methods will produce negative water depth.In order to overcome the above problems,we first rewrite the Green-Naghdi equations into a coupled system of balance laws and an elliptic equation,thus eliminating the mixed derivatives of time and space in flux and source terms.Then,a well-balanced central discontinuous Galerkin finite element method and a positivity-preserving(non-negative)central discontinuous Galerkin-finite element method are proposed respectively.The former is used to maintain the balance between flux and source terms,and the latter is used to maintain the non-negative of water depth.Finally,we propose a positivity-preserving well-balanced central discontinuous Galerkin-finite element method,which can keep the balance between flux and source terms as well as nonnegativity of water depth simultaneously.There are still two problems in solving the coupled system of two-dimensional nonlinear shallow water wave equations and sediment transport equations.One is the problem of still-water stationary solution,the other is the non-negative problem of volume sediment concentration.Therefore,we first propose a well-balanced central discontinuous Galerkin method to keep the balance between the flux and the source terms,and then propose a positivity-preserving central discontinuous Galerkin method to keep the non-negative volume sediment concentration.When solving the incompressible Navier-Stokes equations with variable density,it is necessary to keep the upper and lower bounds of density,especially the problem of high density ratio.In order to design this kind of high-order accurate numerical schemes,the high-order discontinuous Galerkin method or the finite volume method with a bound preserving limiter can be used to discretize the density equation,and any other popular numerical methods can be used to discretize the momentum equation and the pressure equation.We will use a combination of the discontinuous Galerkin method and the finite element method.Specifically,we will use the discontinuous Galerkin method to discretize the density equation,and the finite element method to discretize the momentum equation and the pressure equation.