Parallel Iterative Solution Of Partial Differential Equations And Mesh Optimization

There is an endless demand for the scale of engineering simulation. With the high speed development of computers, the problem size within the capability of computers ranues from thousand to million or even to gillion. The algorithms evolve from serial algorithms to parallel algorithms adapting to large scale parallel computers. When the size of the problem increases, some new problems anse. The first is the parallel structure asks for new algorithms to explore the potential ability, which pushes the study of parallel algorithms forward. The second is that the large scale computing requires more strict stability, which has activated the study of the stability of algorithms and the mesh optimization for ALE simulation. Based on these observations, the main target of this dissertation is the algorithm development and analysis for the parallel simulation of partial differentia! equations.In the aspect of the structure of parallel computers. the large scale distributed memory parallel computers (MPP and PC cluster) lead the trend, which is characterized by the distributed memory structure. The global communication and synchronization are the bottlenecks of large scale numerical simulations. Global communication is a global consideration of the whole problem, which may be inevitable. But it may be replaced In some other techniques. A main problem faced in many engineering applications is the solution of large scale linear systems. It is a very important problem to solve the large scale ill-conditioned linear systems efficiently. The main approach is to make full use of the background of the linear systems to improve the performance of the solver.The studv should penetrate the algebra realm to the background of the theory of PDEs, and even to physical problems. The finite difference schemes for PDEs are very important for the discretization, but the importance of parallelism is only realized within the last twenty years. Another important probiem is the combination of the construction of difference schemes and the solution of the resulted algebra linear systems. Actually, the success of difference schemes tightly depends on the solution of linear systems. Only considering all these factors can a practical scheme be constructed. Another problem of large scaie computing is when the mesh is fine for the sake of precision for high speed fluid, the computation may break down because of the fold of the meshes. It has been a serious problem for high resolution simulation of hard problems.For these problems, this dissertation studies several aspects of the difference solution of the partial differential equations. It gives a detailed discussion on how to make use of massive parallel computers, how to solve PDE stably and efficiently. The dissertation is characterized by combining iterative methods and difference scheme together, which yields a new direction for the solution of radiation hydrodynamics and particle transport problem. Some new parallel difference methods are developed and some theoretical results are analyzed. The idea is applied for parabolic equations and transport equations. The red-black hybrid method is studied in detail, and the acceleration and convergence rate estimation are obtained. The method is applied to radiation hydrodynamics problem and oil simulation problems and result in good performance. Activated by the idea of parallel difference schemes, a parallel preconditioner is developed. And the spectral analysis is carried out for it. And a mesh optimization method is developed for the ALE simulation.The total dissertation can he roughly divided into three parts, i.e. the study of the parallel difference schemes, the effective iterative solution of linear algebra systems and the mesh optimization. The three parts affect each oilier. The structure of the dissertation is as follows. The first is the introduction, which introduce the background and the summary of this dissertation. The first, second and third chapters are dedicated to the study of parallel difference scheme for parabolic type equat...