Adaptive Discontinuous Finite Element Method For Solving The Three Dimensional Euler Equations

In many practical engineering problems, we need to solve differential equations. However, solving differential equations requires a large quantity of calculations. Especially in dealing with three-dimensional problems, one usually uses the very fine computational mesh in order to control the error. Thus, the total elements number will be increased, which will require a large amount of computational resources both CPU time and memory. Actually, the discontinuities only localize in small regions in many cases, while in most regions the solution is smooth. If the uniform grid refinement is adopted, the computational cost would be tremendous. Hence, in order to solve this kind of differential equations in efficient manner, one needs the adaptive methods. The basic goal of the adaptive methods is that the grid can adapt dynamically to reflect local changes of the solution. And it needn't manually set up or modify the grid any longer so that it can save a lot of computational time.The discontinuous Galerkin method (DG) has many advantages, such as local conservation, easy handling to complicated geometries, allowance to discontinuities, formal high order accuracy, and easy realization of parallelization and h-p adaptive calculation. These features make it very suitable to solve fluid dynamic equations. In this thesis, we combine the Runge-Kutta discontinuous finite element method (RKDG) with that adaptive method to solve Euler equations. And the numerical experiments confirm the efficiency of the schemes.In chapter one, it briefly introduces the development of discontinuous Galerkin method. And it describes the background and application of adaptive methods, gives the basic idea of adaptive method and its classification.In chapter two, the adaptive discontinuous Galerkin method is applied to solve the Euler equations on conforming unstructured tetrahedral mesh. We utilize the Alberta package to generate computational mesh, which it makes us a great convenience. Meanwhile, we give four kinds of adaptive strategies and the principle of data transmission. Finally, several numerical examples show the validity of the method. In chapter three, since the transition layer from coarse mesh to fine mesh is wide for the adaptive mesh refinement on conforming mesh, the adaptive discontinuous Galerkin method is applied on the nonconforming tetrahedral mesh. An effective way is developed to make clear the relationship between one element and its adjacent elements. Furthermore, we give three different adaptive strategies, principle of data transmission and error indicator. At last, numerical examples show our original design intent.In chapter four, we discuss the adaptive discontinuous Galerkin method on time, namely local time stepping (LTS). We modify the high order Riemann solver (HEOC) so that it can be applied in the numerical flux calculation. Then, we employ the LTS technology in discontinuous Galerkin method. In the numerical experiments, the finite element space is taken as the quadratic polynomials. By comparing different grid scales, the results verify the validity of the method.