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Parallel Computation Based on Domain Decompositin Method for Parabolic Equation

Author: HeCaiFeng
Tutor: LiuBo
School: Jilin University
Course: Computational Mathematics
Keywords: Domain Decomposition D-N Alternating Process Parallel Compu-tation Stability Truncation Error
CLC: O241.82
Type: Master's thesis
Year: 2011
Downloads: 46
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Abstract

In many areas of natural science, a lot of phenomena is describing with parabolic equation or equations. Such as heat conduction or other diffusion phenomenon,chemical reaction, certain Biological form, all kinds of particle transport etc. Because many problems have large scale, the research of us ing finite difference methods to solve parabolic partial differential equation numerically has important theoretical significance and application value.The classical explicit method possesses an ideal parallelism, which is very suitable for parallel computing, but it has a conditional stability, especially in the multidimensional problems, the time step of computation is restricted strictly. The classical implicit method and Crank-Niclson format are both absolutely stable, but they include a process of solving banded equations. So the classical implicit method and Crank-Niclson format can’t be applied on parallel machine directly and effectively. Therefore, we need to develop some new difference methods with good stability, parallelism and precision. In the 1980s, Evans and Abdullah’s designed an method called alternating group explicit (AGE) method, which not only ensures the stability of numerical calculation, but has good parallel nature thanks to the explicit solution. This achievement indicates that it is possible to design a new difference format with good stability, parallelism and precision.Domain decomposition method is a numerical solution method for par-tial differential equations with a high degree of parallelism. It’s basic idea is to decomposing the solution domain into some small and regular subdomains, and then the solution of the original equation is transformed into the solu-tion of some equations on the subdomains. This algorithm is highly parallel, because the main steps is calculated in each subdomain independently.In this paper we develop a new parallel difference arithmetic for solving parabolic equations based upon the idea of D-N alternating iterative method, which is one of the non-overlapping domain decomposition methods used for solving elliptic equations. The new algorithm is a discrete domain decompo-sition algorithm for parabolic equations based on the grid decomposition. We divide the grid domain into some subdomains and then establish the corre-sponding discrete difference format on each subdomain. These formats are independent, can be computing simultaneously. The proof of the new algo-rithm’s stability and it’s truncation errors analysis are given by means of the previous theory of finite difference parallel method. Finally the numeric ex-periment shows that the new algorithm presented in this paper is effective.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > The numerical solution of differential equations, integral equations > Numerical Solution of Partial Differential Equations
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