Moving Messh Method And Its Applications Coupled With Dynamic Domain Decomposition

Adaptive moving mesh method has become an effective numerical method for solv-ing problems, of which the solutions evolve dynamically singular or near singular in fairly localized regions. And with the help of domain decomposition methods, we can not only accelerate the problem solving by parallel computing, but also apply different physical models in different regions, so that the coupling model can describe the physical phenom-ena more realistic but not increase the computational complexity. This thesis is mainly concerned with the coupling of moving mesh method and dynamic domain decomposi-tion. In particular, we focus on the generation of moving mesh based on dynamic domain decomposition, the moving mesh method for the dynamic coupling model, and the ap-plications to the problems such as moving interfaces problems, traveling singular sources problems, the numerical simulations of rarefied gas, etc.We first consider the moving mesh method for moving boundary problems, which is the basis for the generation of moving mesh based on dynamic domain decomposition. Then we study the moving mesh method for moving interfaces problems and traveling singular sources problems. According to the locations of interfaces or singular sources, we dynamically divide the domain into several subdomains, and apply moving mesh strategy on each subdomain independently to obtain the local adaptive mesh on it. The physical PDE is solved on each subdomain with its local mesh independently, or on the whole domain with the global mesh composed of the local mesh of each subdomain. Benefit from the domain decomposition technique, the moving mesh equation could be solved quickly, which further improves the efficiency of the moving mesh method. Moreover, the global mesh consisting of the local mesh always satisfies that there exists a fixed index ji, such that the ji,th mesh point just locates at the ith moving interface or singular source. Taking this advantage, the calculation of the jump [u] is avoided, which greatly simplifies the construction of the second-order numerical scheme for the physical PDE, and also provides an idea for numerical method of high dimensional problems. In addition, the algorithms proposed in this thesis can be directly applied to the cases of multi-interfaces or multi-sources. Numerical examples are presented to verify the validity of our algorithms, and illustrate their efficiency in solving singular problems, especially blow-up problems. By these algorithms, we also investigate the blow-up phenomena for traveling singular sources problems with various motions of the sources.For applications of moving mesh method to the dynamic coupling model, we take the kinetic/hydrodynamic coupling model for the numerical simulation of rarefied gas as an example. We utilize the moving mesh framework based on harmonic maps, which is pri-marily split into three parts:the numerical method on static mesh for the underlying PDE, the algorithm for solving the mesh PDEs, and an interpolation algorithm for numerical so-lutions between the old mesh and the new mesh. The key ingredients of this moving mesh method are the monitor function for the mesh PDEs and the interpolation algorithm. Based on the numerical solutions of the coupling model, we design a mesh-independent monitor function, so that mesh points can cluster to areas with singularities or large solution varia-tions, which always contain the kinetic regions. Since solutions for the coupling model are only known in each regions, it is difficult and inefficient to interpolate them on the whole domain. And another problem is for the mesh grids moving from the hydrodynamic re-gions into other regions, we still do not have the numerical solutions for the kinetic model to interpolate, but these are necessary for the evolution of the underlying PDE in the next time step. To this end, we reuse the cut-off function, which identifies the regions for each model of the coupling, so that we can only update the solutions in the necessary areas. We also derive a conservative interpolation scheme for numerical solutions. Finally, numerical tests are given to show well performs of our moving mesh method in deal with the complex physical models, which demonstrates the validity and the efficiency of our method.