The Development And Application Of Central Discontinuous Galerkin Method

Central discontinuous Galerkin method is a class of finite element methods for con servation laws.By evolving two sets of numerical solutions defined on overlapping mes hes,the central discontinuous Galerkin method method does not rely on any exact or ap proximate Riemann solver at element interfaces as in discontinuous Galerkin method.T herefore,the central discontinuous Galerkin method is widely applied to various proble ms.This dissertation consists of two parts.The first part concerns the development of the central discontinuous Galerkin method on unstructured overlapping meshes and its applications in various conservation laws.The second part is devoted to the designing and analysis of the efficient central discontinuous Galerkin method and its applications in various conservation laws.The existing central discontinuous Galerkin methods for conservation laws are defined on structured overlapping meshes.Therefore,the methods are only applied to problems defined on simple domains,such as rectangle,L-shape domain.In order to extend the methods to more complex domains,we present a family of high order central discontinuous Galerkin methods defined on unstructured overlapping meshes for the two dimensional conservation laws.The primal mesh is a triangulation of the computational domain,while the dual mesh is a quadrangular partition which is formed by connecting an interior point and the three vertexes of each triangle in the primal mesh.We prove the stability of the present method for linear equation.Then,the proposed central discontinuous Galerkin method on unstructured overlapping meshes is employed to solve the ideal magnetohydrodynamic equations.We also propose and numerically investigate a family of locally divergence-free central discontinuous Galerkin methods for ideal magnetohydrodynamic equations.The performance of the proposed methods will be demonstrated through a set of numerical experiments.Central discontinuous Galerkin methods are a family of high order numerical methods,which evolve two sets of numerical solutions defined on overlapping meshes and do not calculate the numerical flux at element interfaces as in discontinuous Galerkin methods.However,evolving two sets of numerical solutions makes the central discontinuous Galerkin method time-consuming.In this dissertation,we present a reconstructed central discontinuous Galerkin method for conservation laws.In this scheme,we reconstruct an approximate solution by projecting the numerical solution defined on the primal mesh into the approximate space defined on the dual mesh.The reconstructed approximate solution is used to replace the numerical solution defined on the dual mesh in the central discontinuous Galerkin method.The reconstructed central discontinuous Galerkin method is simpler and easier to implement than the central discontinuous Galerkin method.We also study the stability and error estimate for smooth solutions of the reconstructed central discontinuous Galerkin method for linear hyperbolic equation.Then,the proposed reconstructed central discontinuous Galerkin method is applied to the fully nonlinear weakly dispersive Green-Naghdi model.We present a class of high order reconstructed central discontinuous Galerkin-finite element methods for the fully nonlinear weakly dispersive Green-Naghdi model.The proposed methods reduce the computational cost of the traditional methods by nearly half but still maintain the formal high order accuracy.Numerical tests are presented to illustrate the accuracy and computational efficiency of the proposed method.