Structure Preserving Discontinuous Galerkin Methods For Hyperbolic Conservation Law With Source Term

In this thesis,we develop the structure-preserving discontinuous Galerkin(DG)and arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods for a class of hyperbolic conservation laws with source term,which can preserve a general hydrostatic equilibrium state and positivity-preserving property under a suitable time step at the same time.Such equations mainly include the shallow water equations with non-flat bottom topography and the Euler equations with gravitation.By introducing well-balanced numerical fluxes and corresponding source term approximations,we established well-balanced schemes.We also discuss the weak positivity property of our proposed schemes,and the positivity-preserving limiter can be applied to effectively enforce the positivity-preserving property.This thesis is mainly divided into three parts.In the first part,we develop the well-balanced positivity-preserving DG methods for the Euler equations with gravitation on unstructured mesh.The main efforts in this work are two parts.The first part is a class of special cell boundary values and corresponding source term approximation,which are the key to the robustness of the scheme and can make the well-balanced scheme have better performance in the accuracy test.The design of such cell boundary values should maintain the fluid velocity and keep the fluid density and pressure greater than zero.Such design greatly simplifies the proof of the weak positivity property of the scheme.The second part is a novel projection operator for the recovery of the reference equilibrium state.Such projection simplifies the wellbalanced scheme and the corresponding proof of the weak positivity property.At the same time,the newly defined projection operator can naturally extend the well-balanced scheme to two-dimensional unstructured mesh to deal with the complicated or irregular computational domains.We carry out rigorous proof for the well-balanced property and the positivity-preserving property of our proposed scheme.Numerical examples have been provided to demonstrate the good properties of our scheme.In the second part,we develop and analyze the well-balanced positivity-preserving ALE-DG methods for the shallow water wave equations with non-flat bottom topography on moving mesh.We design the well-balanced schemes for two equilibrium states of the shallow water equations,the still water equilibrium and the moving water equilibrium,which have wide applications.All proposed schemes satisfy the positivitypreserving property.For the still water equilibrium,the mass conservation and the error caused by the time discretization on moving meshes make it difficult for ALE-DG methods to maintain the well-balanced property.We first change the main variables of the shallow water equations and then use the geometric conservation law(GCL)of the ALE-DG methods to construct a well-balanced scheme.Considering the mass conservation and the positivity-preserving property,we introduce a novel approximation based on the ALE-DG methods for the bottom topography.Two different well-balanced ALE-DG schemes have been proposed based on the different well-balanced techniques on static mesh and rigorous proofs have been carried out for the well-balanced property and the positivity-preserving property.For the moving water equilibrium,we assume that the desired equilibrium state has been explicit given and define its approximation based on the ALE-DG methods.We reasonably define the well-balanced property on moving mesh and carefully define the well-balanced numerical fluxes to maintain the mass conservation and the positivity-preserving property.A large number of numerical examples have been provided to demonstrate the properties we have declared and the good performance of our proposed schemes.In the third part,we propose the well-balanced positivity-preserving ALE-DG methods for the Euler equations with gravitation on moving mesh.Considering the mass conservation and the error caused by the time discretization on moving mesh,we assume that the desired equilibrium state has been explicit given and design wellbalanced schemes on such assumption.Considering the positivity-preserving property of the schemes,a novel approximation of the desired well-balanced scheme has been introduced based on the ALE-DG methods.Since the numerical fluxes contain the grid velocity,we carefully design the numerical fluxes to maintain the mass conservation and the positivity-preserving property.Based on two different well-balanced techniques on static mesh,we developed two different well-balanced positivity-preserving ALE-DG schemes.We rigorous proved that both two schemes have the well-balanced property and the weak positivity property.A simple scaling limiter can be applied to ensure the positivity-preserving property.This allows the proposed schemes can preserve the equilibrium states involving low density and low pressure on moving mesh.Numerical examples have been provided not only to demonstrate the good properties but also to show the advantages for the examples with shocks of our schemes on moving mesh.