A Finite Difference Domain Decomposition Algorithm For Parabolic Equation

In many fields of the natural science, many phenomenas are described by parabolic equation or equations. Heat equation is the most typical one of parapolic equations, which describes many physical-phenomenas, such as conduction, diffusion, etc. We are expreriencingincreasing tribulations by using the typical finite difference methods to solve those parabolic-equations. Hence, it is very meaningful to divide the domain over which the problem is defined into subdomains, and solve the subdomain problem in parallel.First, a brief introduction is given to the domain decomposition algorithm. Domain decompositon algorithm is a new effective method developed to solve the partial differentialequation in parallel. Lots of attention is paid to the domain decomposition algorithm for its numerous advantages. Mathematicians from the whole word including America, Russia, France etc. arc interesting in the method and developing their genres. According to their different decomposition of the computation domain, domain decomposition algorithmare divided into un-overlapping domain decomposition algorithm, overlaping domaindecomposition algorithm, dummy domain decomposition algorithm , multi-parallel domain decomposition algorithm etc. Lots of physical and mechanical problems could be ended into the solution of the parabolical equation. Finite difference domain decompositionalgorithm of the parabolical equation is developing rapidly since 1990s. Particular and profound research are given by C.N.Dawson, Qiang Du and T.F.Dupont to the parabolical equation by using domain decomposition algorithm. A Brief introduction of our algorithms for parabolic equation is given at the end. The one space dimensional parabolical equation:u is the solution of the parabolical equation.Based on.the algorithm for the parabolic equation with constant coefficent by C.N.Dawson etc. ,we develop a new algorithm for parabolical equation. The domain decomposition method based on Du Fort- Frankel scheme at the interface point and fully implicit scheme at interior points for the parabolic equation. And the precision of the new algorithmis equal to the precision of the implicit schemes, so we design a new algorithm for the parabolical equation. The stability-condition and precision are well.algorithm:U_iⁿ= u_iⁿ, at boundary points,at interface points, n = 1, 2,…, Mat interior points, n = 0,1,…, M,at interface points, n = 0,In this paper, we'll discuss the algorithm stability and convergence, and also the numerical examples of simulation are given. Some characters are proved by numerical experiments in this paper. By performance analysis of parallel numerical experiments, the method with intrinsic parallelism has the same accurate as the fully implicit scheme, which suit large scale scientific computing and engineering computer.The two space dimensional parabolical equation: u is the solution of the parabolic equation, andwe give a straightforward generalization of the one-dimensional results to two space dimensions, get a new algorithm:algorithm:U_i,jⁿ= u_i,jⁿ, at boundary points,at interface points, n = 1,2,…, Mat interior points, n = 0,1,…, M,at interface points, n = 0,we also discussed the algorithm stability and convergence, we can draw the conclusionthat the algorithm for two space dimensional parabolical equation suit large scale scientific computing and engineering computing.This thesis will be divided into 4 chapters, organized as follows:In chapter 1, the developments of the domain decomposition algorithm and finite differencedomain decomposition algorithm for numerical solution of the parabolical equation have been introduced, and the main work of this paper is also described.In chapter 2, first we introduce the difference domain decomposition algorithm for parabolic equation with constant coemcent at interface points designed by C.N.Dawson and Sheng Zhiqiang etc. and the error estimate results. In this paper we develop a new algorithm for the parabolic equation , the domain decomposition method Based on Du Fort - Frankel scheme at the interface point and fully implicit scheme at interior points for the parabolic equation. And the precision of the new algorithm is equal to the precisionof the implicit schemes, so we design a new algorithm for the parabolical equation. The stability-condition and precision are well. Hence we can use a larger time step, which can save a lot of computational works for the parallel solution of the parabolic problem. And the new algorithm suit large scale scientific computing and engineering computing.In chapter 3. we discussed the two space dimensional parabolical equation. We give a straightforward generalization of the one-dimensional results to two space dimensions, get a new algorithm for two space dimensional parabolical equation, and we prove the stability and convergence of the new algorithm for the two space dimensional parabolical equation.In chapter 4, we give a numerical example and numerical results to validate the algorithm.