The Study Of Discontinuous Galerkin Methods With Property Preserving

In this thesis,we mainly study:the globally divergence-free discontinuous Galerkin(DG)methods for ideal magnetohydrodynamic(MHD)equations,the maximum princi-ple satisfying arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method for scalar conservation laws and the positivity preserving ALE-DG methods for com-pressible Euler equations,and discontinuous Galerkin methods with optimal L2 accu-racy for partial difference equations with high order spatial derivatives.In the first part,we propose one globally divergence-free numerical method to solve the ideal MHD equations.The ideal MHD equations can be decomposed in two systems.Our algorithm is based on discontinuous Galerkin methods in space.First,we discrete the induction equation separately to approximate the normal components of the magnetic field on element interfaces.We utilize DG methods to solve nonlinear systems for fluid variables.Based on H(div)finite element space,we use an element by element reconstruction to obtain the globally divergence-free magnetic field.In time,we utilize the strong-stability-preserving Runge-Kutta methods.In consideration of accuracy and stability of the methods,a careful investigation is carried out,both nu-merically and analytically,to study the choices of the numerical fluxes associated with the electric field at element interfaces and vertices.The resulting methods are local and the approximated magnetic fields are globally divergence-free.Numerical examples are presented to demonstrate the accuracy and robustness of the methods.In the second part,we develop the maximum principle satisfying ALE-DG meth-ods to solve scalar conservation laws on two dimensional simplex meshes and the pos-itivity preserving ALE-DG methods to solve the compressible Euler equations in one dimension.We first propose the semi-discrete ALE-DG methods and analyze its L2 sta-bility.For scalar conservation laws,we develop the fully-discrete ALE-DG method on two dimensional simplex meshes.The relationship between the spatial dimension and the geometric conservation law is elaborated,and we prove our ALE-DG method satis-fies the geometric conservation law,when the order of any time discretization methods is greater than or equal to the dimension of space.In addition,when we use the bound preserving limiter proposed by Zhang,Xia and Shu(J.Sci.Comput.50(2012),29-62),we prove that our fully discrete ALE-DG method satisfies maximum principle.Based on the ALE-DG methods for scalar conservation laws,we construct a positivity pre-serving ALE-DG method for solving one-dimensional compressible Euler equations.We prove that the cell average value of approximations for density and pressure from the ALE-DG methods are positive under a certain condition for CFL number.Then,we apply the positivity preserving limiter developed by Zhang and Shu(JCP,229(23),2010,8918-8934)to our ALE-DG method,and obtain positive density and pressure from high order ALE-DG methods for Euler equations.The numerical results on two dimensional simplex meshes are demonstrated to verify our theoretical results of ALE-DG methods for scalar conservation laws,and the one dimensional numerical results show the numerical stability,robustness of our positivity preserving ALE-DG methods for compressible Euler problems with low density or low pressure.In the third part,we formulate and analyze discontinuous Galerkin(DG)methods with L2 optimal accuracy to solve several partial differential equations with high or-der spatial derivatives in one dimension.These equations include the heat equation,a third order wave equation,a fourth order equation and the linear Schr(?)dinger equation.Based on auxiliary variables,we first rewrite each PDE into its first order form and then apply a general DG formulation.Then,we design one group of numerical fluxes as linear combinations of average values of fluxes,and jumps of the solution as well as the auxiliary variables at cell interfaces.The main focus of this part is to identify a sub-family of the numerical fluxes by choosing the coefficients in the linear combinations,so the solution and some auxiliary variables of the proposed DG methods are optimally accurate in the L2 norm.In error estimates,we develop some special projection opera-tor(s),tailored for each choice of numerical fluxes in the sub-family,to eliminate those terms at cell interfaces that would otherwise contribute to the sub-optimality of the error estimates.Our theoretical findings are validated by a set of numerical examples.