Finite Difference Domain Decomposition Algorithm And Alternating Direct Implicit Method For Parabolic Problems

Mathematical physics and engineering problems, such as the aerodynam-ics, the construction of large water conservancy facilities, the exploration and exploitation of oil and gas reservoir, the design of spacecraft, could all attribute to the solutions of high-dimensional large partial differential equation models. The large scale of computing and irregularity of computational domain bring difficulties for the scientific computing. However,higher and higher computing precision is needed to solve these problems and the current computing speed of computers could no longer meet the actual computing demand. Thus as the parallel computing is making significant progress, domain decomposition method is becoming an effective solution to the partial differential equations and is well used on many fields.Shortly speaking, domain decomposition method divides the whole do-main into several sub-domains, and the shape of these sub-domains may as well be regular. And then the solutions of the original problems can be translat-ed into solving the problems on the sub-domains respectively. Domain decom-position methods have many advantages that other algorithms do not have: first, it can reduce the computing scale; second,the computing on the sub-domain can be parallel, thus the computing time is shortened; third, if the shapes on the sun-domains are regular enough, we can adopt the common familiar fast algorithms or the existent efficient softwares to solve the prob- lems,and then the workload can be reduced.The author has finished the dissertation based on the above researchers’ studies, which solve some work on domain decomposition methods. The pa-per is divided into four chapters.The first chapter is "Introduction" section, we mainly make some preliminary knowledge of domain decomposition algo-rithm and the finite difference algorithm and summarize the above researchers’ studies. In chapter two, we use domain decomposition methods for solving one-dimensional heat conduction problem.We export Saul’ yew symmetric implicit format based on classic explicit format and get the error estimates. Numeri-cal examples show that the error of algorithm is less than the classic implicit format. In chapter three, we use the alternating direction method for solv-ing two-dimensional heat conduction equation. First we export higher-order difference format on one-dimensional equation, then by using an intermediate layer, we get the high-order alternating direction difference format,at last,we prove the unconditional stability of this format. Numerical examples show the format can indeed achieve O(Δt2+h4). The final chapter get the numerical solution of two-dimensional parabolic equation, using the domain decompo-sition method, based on alternating directions methods in the chapter three. This chapter introduced divide the initial area into sub-domains, and then use iterative method to solve the value of the sub-domains, we get the value of the initial area by splitting the value of the subdomains.At last, we get the error estimates.