The Parallel Difference Scheme For Convection-diffusion Equation

Convection-diffusion equations are important partial differential equations. They are a kind of basic motion equations. They can be used to describe the distribution of pollutants in the river pollution, air pollution and nuclear waste pollution. They also can be used to describe the fluid flow, the heat conduction in the fluid and many other physical phenomenons. So, the computation of the convection diffusion equations has very important significance in theory and practice. There are many numerical methods for convection-diffusion equations. The finite difference method is one important numerical method for fluid dynamics.Along with the development of the high-performance computers, parallel algorithms become more and more important. They have been developed several decades, and got great success. Especially, there are many achievements in recent twenty years, D.J. Evans, A. R. B. Abdullah, Yulin Zhou, Baolin Zhang, Guangwei Yuan and many other mathematicians have done a lot of hard work.In this thesis, we propose some parallel difference schemes for diffusion equation and convection-diffusion equation. First, in Chapter two a new alternating segment explicit-implicit (ASE-I) method with exponential type for the numerical solution of the convection-diffusion equation is derived. The method has the obvious property of parallelism, and is unconditionally stable. Numerical example is given which illustrates that the present method is better than the classical ASE-I method for solving the convection-diffusion equation. Second, in Chapter three we construct a new difference scheme, and a weighted AGEI method for solving the convection-diffusion equation is derived on the basis of the new scheme. Numerical experiments show that this method is effective. Third, because of the similarity, we can use the similar method to solve Burgers equation. Numerical experiments show that it is a useful method to solve Burgers equation. Then, this paper also studies the new domain decomposition algorithm for two-dimensional heat conduction equation. In the calculation of the interior points, we use the full implicit scheme. In the calculation of the interface points, we use a new scheme with a fractional step larger spacing. Using this method, the stability bound is released by 2m2q for the one dimensional parabolic problem and 4m2q for the two dimensional parabolic problem respectively. Finally, we carry out some parallel difference schemes on the parallel machine. In each chapter, we give some numerical experiments to test the schemes proposed. The numerical results confirm the theoretical predications, and show that these schemes are robust and effective.