Solving The3-D Compressible Euler Equations By Using The Adaptive Discontinued Galerkin Method

The partial differential equations are widely applied in atmospheric physics, astrophysics, combustion and explosion theory, aviation and aerospace, inertial confinement fusion, oil exploration, and so on. The Euler equations and radiation diffusion equations have a strong physical meaning and are main research contents of radiation hydrodynamics. How effective and efficient for solving such nonlinear problems for the majority of researchers is an important to explore topic.The adaptive discontinuous finite element method and preconditioned Jacobian-free Newton-Krylov method are very suitable to solve some large-scale, multi-scale and highly nonlinear problems. They have small amounts of calculation. So, the two kinds of methods are widely applied in the fields of fluid dynamics and heat transfer. In this article, the adaptive discontinuous finite element method and the preconditioned Jacobian-free Newton-Krylov method are used to solve nonlinear problems by efficiently designing the reasonable algorithms, and numerical experiments demonstrate the efficiency of our method.In Chapter1, several discontinuous finite methods are introduced which include the Runge-Kutta discontinuous finite element method (RKDG) and the Lax-Wendroff time discrete discontinuous finite element method (LWDG) and the local discontinuous finite element method (LDG). In the following, three kinds of adaptive methods are introduced. Finally, the inexact Newton method and Krylov subspace methods are introduced including the GMRES, BiCGSTAB and TFQMR method. We analyze the relationship of the three methods, and compare their advantages and disadvantages with each other.In Chapter2, the adaptive LWDG method is applied to solve the three-dimension-al hyperbolic conservation laws. The effective adaptive strategie is given and an equidistribution compromise strategy is carried out on the nonconforming tetrahedral meshes. The two dimensional case of the posteriori error indicators is generalized to three-dimensional hyperbolic conservation laws. Compared with the traditional second order RKDG method, the LWDG method has a smaller calculation amount and higher order accuracy. Numerical experiments demonstrate the efficiency of our method.In Chapter3, the adaptive mesh refinement (AMR) discontinuous finite element method is applied to solve the three-dimensional compressible Euler equations on nonconforming tetrahedral mesh. The mesh refinement or coarsening is determined according to the error indicators given by the gradients of the different physical quantities. For three-dimensional problems, the density, energy, pressure and enthalpy error indicators are compared. Numerical results show their different effectiveness.In Chapter4, the reconstruction adaptive discontinuous finite element method is studied on conforming meshes. By expanding into the DG solution the Taylor series on the centers of tetrahedral meshes and using the discontinuous finite element method to calculate the average of the values and the first derivative values on the current element and its adjacent elements, the high accuracy numerical solution of the three-dimensional compressible Euler equations is obtained. Compared with the second order adaptive RKDG method, the reconstruction adaptive discontinuous finite element method has a smaller amount of calculation.In Chapter5, the local time stepping adaptive discontinuous finite element method is applied to solve the three-dimensional compressible Euler equations. The improved Osher NC is applied to numerical flux function calculations. Comparing with the uniform Runge-Kutta discontinuous finite element method, the local time stepping adaptive discontinuous finite element method is of distinguishing feature with smaller amount of calculations.In the final chapter, the Jacobian-free Newton-Krylov (JFNK) method is applied to solve large scale nonlinear equations. One of the most important advantages of the JFNK method is that it needn’t form and store the Jacobian matrix of the nonlinear system. To apply preconditioning technique to solve a class of non-equilibrium radiation diffusion problems coupled with material thermal equations, we design two kinds of linearizing methods for preconditioners. Numerical results show that these two preconditioning schemes can improve the convergence of the JFNK method.