The Research Of Discontinuous Galerkin Method For Ideal Magnetohydrodynamics Equations Under The Lagrangian Framework

Magnetohydrodynamics(MHD)equations are the most basic model which describes how the magnetic field interacts with conducting fluid.They are ap-plied in a very wide range of scientific researches such as astrophysics,controlled thermonuclear fusion and metal smelting,etc.Some of these problems are hard-ly to carry out experimental observation,therefore,it is of great importance to design accurate and robust numerical methods for the MHD equations.In this thesis.We adopt the Discontinuous Galerkin(DG)method to research the ideal MHD equations in one and two dimensions,design several numerical schemes and obtain some comparatively ideal numerical results.First,we present a Runge-Kutta Discontinuous Galerkin(RKDG)method for two dimensional MHD equations on quadrilateral meshes under the La-grangian framework.To deal with the divergence-free constraint condition of the magnetic field,we adopt the local divergence-free function space to approx-imate the magnetic field.A HWENO reconstruction limiter is designed for the numerical solution to prevent the non-physical oscillation for flows with strong discontinuities.We prove theoretically that our limiter can not only preserve the local divergence-free property of the magnetic field,but also avoid to calculate the complex eigen-system of the MHD equations.Some numerical examples are presented to demonstrate the accuracy and the non-oscillatory property of the scheme.According to the physical principle that there are no magnetic monopoles,therefore,the magnetic field must satisfied the divergence-free constraint.How-ever,the above numerical scheme can only obtain the local divergence-free numer-ical magnetic field.To ensure that the numerical magnetic field possesses global divergence-free property,we develop a RKDG method which preserves the ex-actly divergence-free property of the magnetic field to solve the two-dimensional Lagrangian ideal MHD equations.In this method,fluid part of the ideal MHD equations and the magnetic induction equation along with z-component are discretized by the RKDG method.The magnetic field discretizition in the x-and y-direction are obtained by constrcuting through the magnetic flux-freezing principle.Since the divergence of the magnetic field in 2D is independent on its z-direction component,an exactly divergence-free numerical magnetic field can be obtained by this treatment.Some numerical examples are presented to demonstrate the accuracy,non-oscillatory property and preserving the exactly divergence-free property of our method.In the physical phenomenon which described by the ideal MHD equations,the density and thermal pressure are always non-negative physical quantities.However,such positivity property is not always satisfied by approximate solu-tions of numerical schemes.In order to handle this problem,we design a La-grangian HLLD approximate Riemann solver so that a first order Lagrangian DG scheme can preserve the positivity of density and thermal pressure.Then we develop a high order positivity-preserving Lagrangian DG scheme for solving one-dimensional ideal MHD equations by using the strong stability preserving(SSP)high order time discretizations and the positivity-preserving scaling lim-iter.Some numerical examples are presented to demonstrate the accuracy and positivity-preserving property of our scheme.